A lamémoire dechemori/Temp/Auwal/Thesis/Thesis_Maalouf... · 2013-09-30 · 6.5 Design of a multi-variable controller for depth and pitch control in under- ... According to safety

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Contents

Contents iii

General introduction 1

I Introduction and State of the Art 3

1 Context and problem formulation 5

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Assisting the pilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Main contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 State of the art on control schemes 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Classification of main existing control schemes . . . . . . . . . . . . . . . . . . 16

2.3 Classical control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Robust control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Adaptive control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Intelligent control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Hybrid control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8 Comparison between the various schemes . . . . . . . . . . . . . . . . . . . . 32

2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

iii

iv CONTENTS

3 Modeling of underwater vehicles 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Thruster dynamic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.1 Propeller shaft speed models . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.2 Thrust modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

II Proposed Solutions 47

4 Solution 1: Conventional controllers 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 PID control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.2 PID Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.3 Application for depth and pitch control . . . . . . . . . . . . . . . . . . 53

4.3 Nonlinear adaptive state feedback control . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.2 Application for depth and pitch control . . . . . . . . . . . . . . . . . . 55

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Solution 2: Nonlinear L1 adaptive controller 57

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 From MRAC to L1 adaptive control . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2.1 From direct MRAC to direct MRAC with state predictor . . . . . . . . . 59

5.2.2 From direct MRAC with state predictor to L1 adaptive control . . . . 61

5.3 Background on L1 adaptive control . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3.1 State feedback L1 controller for linear time invariant systems . . . . . 63

5.4 State feedback L1 controller from nonlinear multi-input systems with un-

certain input gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Design of a multi-variable controller for depth and pitch control in under-

water robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Solution 3: A New Extension of L1 adaptive control 73

CONTENTS v

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Limitation of the original L1 adaptive controller . . . . . . . . . . . . . . . . . 74

6.3 Proposed extension of the L1 adaptive control . . . . . . . . . . . . . . . . . . 75

6.3.1 First variant: a PID based extension . . . . . . . . . . . . . . . . . . . . 75

6.3.2 Second variant: a nonlinear proportional based extension . . . . . . . 76

6.3.3 Validation in simulation on an illustrative example . . . . . . . . . . . 77

6.4 Stability analysis of the extended L1 adaptive control . . . . . . . . . . . . . . 78

6.4.1 Illustrative example for the stability analysis . . . . . . . . . . . . . . . 78

6.4.2 Comparison between the original and the PID based extended L1

adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.4.3 Effects of the PID feedback gains on the stability . . . . . . . . . . . . 80

6.4.4 Effects of the adaptation gain on the stability . . . . . . . . . . . . . . 81

6.5 Design of a multi-variable controller for depth and pitch control in under-

water robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

III Experimental Results 87

7 Experimental case study: the AC-ROV underwater vehicle 89

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2 General features of the AC-ROV vehicle . . . . . . . . . . . . . . . . . . . . . . 90

7.3 Thrusters’ configuration and characteristics . . . . . . . . . . . . . . . . . . . . 91

7.3.1 Thrusters’ configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.3.2 Thrusters’ characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.4 Hardware architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8 Experimental results of the proposed control schemes 97

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.2 Description of the investigated experimental scenarios . . . . . . . . . . . . . 98

8.3 Application of the PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . 99

8.3.1 Controller’s parameters tuning . . . . . . . . . . . . . . . . . . . . . . . 99

8.3.2 Real-time experimental results . . . . . . . . . . . . . . . . . . . . . . . 100

8.4 Application of the NASF controller . . . . . . . . . . . . . . . . . . . . . . . . . 103

8.4.1 Controllers’ parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 103


8.5 Application of the L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . 109

vi CONTENTS



8.6 Application of the extended L1 adaptive controller . . . . . . . . . . . . . . . 114



8.7 Comparison among the various proposed controllers . . . . . . . . . . . . . . 118

8.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

General Conclusion and Perspectives 129

Summary of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A Roll stabilization with an internal rotating disk 133

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A.3 Dynamic Modeling of the Underwater Vehicle . . . . . . . . . . . . . . . . . . 135

A.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

A.3.2 Disturbance effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

A.4 Proposed Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A.4.1 Nonlinear State Feedback Control . . . . . . . . . . . . . . . . . . . . . 138

A.4.2 Roll Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A.4.3 Feedforward for Pitch and Yaw . . . . . . . . . . . . . . . . . . . . . . . 139

A.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A.5.1 Scenario 1: Nonlinear State Feedback applied on the yaw and pitch . 141

A.5.2 Scenario 2: Nonlinear State Feedback applied on the yaw and pitch

with disk-based roll stabilization . . . . . . . . . . . . . . . . . . . . . . 142

A.5.3 Scenario 3: Proposed Control Scheme . . . . . . . . . . . . . . . . . . . 142

A.5.4 Scenario 4: Gyroscopic effects and disk size . . . . . . . . . . . . . . . 143

A.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

B Proof of stability of the NASF 147

C Proof of stability of the AC-ROV with the L1 adaptive controller 149

D Useful Mathematical Tools 157

D.1 Infinity Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

D.1.1 Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

D.1.2 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

D.2 L1 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

CONTENTS vii

D.3 Projection Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

E Details of the model’s parameters 159

Bibliography 163

List of Figures 173

List of Tables 178

General introduction

Ocean depths are until today considered to be a highly unexplored domain since they

have been an unrevealed mystery for centuries. During the past decades, technology and

research witnessed an increased interest in ocean exploration. This need for exploration

gave birth to different types of underwater vehicles amongst which the mini Remotely Op-

erated Vehicles also called mini ROVs.

The use of mini ROVs is covering a big variety of marine activities. Surveillance and

maintenance of subsea installations for instance, can now be made more efficiently and

accurately. However, piloting such vehicles is a tedious task. In fact, given their high power

to weight ratio, these robots are very sensitive to any change in their environment or in

their dynamic model. The addition of an onboard sensor modifies the weight as well as the

hydrodynamic drag of the robot and can affect its performance. An unexpected encoun-

tered obstacle is likely to destabilize the system. Assisting the pilot by partly automatizing

the task to be accomplished helps in reducing time and cost and adds precision to the un-

dertaken mission.

Having established the necessity of automatized or semi-automatized mini ROVs, a

new challenge arises: "How can we make these robots follow a desired trajectory au-

tonomously despite their inherent instability and the disturbances induced by the envi-

ronment". Traditional control schemes often fail to accommodate the inherent nonlinear-

ities of the system under study and achieve the required performance or they require very

fine tuning (depending on the payload and the environment) due to the high sensitivity of

the mini-ROVs. For this reason, the interest in this dissertation has been directed towards

self-tuning methods.

This thesis considers control methods to be designed and implemented on a small-

1

2 GENERAL INTRODUCTION

sized underwater robot. We acknowledge the hazardous unstructured environment in

which the vehicle operates and the highly nonlinear dynamics of the system under study.

The problems considered in the formulation of our control scheme are therefore the un-

certainties underlying the vehicle’s model parameters and their variability, as well as the

disturbances and changes occurring in the operating environment (salinity, mechanical

impacts...). The objective is challenging from a theoretical and practical aspect. In fact, the

methods targeted are advanced and robust in order to cope with a very poor knowledge

of the robot characteristics withstanding experimental conditions possibly encountered

during a designated mission such as waves and random obstacles.

The following chapters will attempt to investigate solutions for the control challenges

and validate them on an experimental platform.

Part I

Introduction and State of the Art

3

CHAPTER

1Context and problem formulation

There is no need to boast of your

accomplishments and what you

can do. A great man is known, he

needs no introduction.

CHERLISA BILES

Contents

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Assisting the pilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Main contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1 Introduction

As seen earlier, underwater vehicles have recently attracted a great deal of interest from

scientists, engineers, industries and control theorists. These various communities envision

in this technology a very useful tool for undersea exploration and complex tasks. Depend-

ing on the mission needed, various types of vehicles can be used. Throughout this chapter,

a closer look on the context conditioning this research will be presented. A description

5

6 CHAPTER 1. CONTEXT AND PROBLEM FORMULATION

(a) AC-ROV (b) Seabotix LBV 300 (c) Ocean Modules V8 Sii

Figure 1.1: Example of mini ROVs used for inspection.(Courtesy of AC-CESS, Seabotix andOcean Modules)

of the underwater vehicles of interest with their applications and challenges will be dis-

cussed. This will therefore lead us to the goal of the thesis and the problem to be tackled.

Finally the chapter will end with the outline of the dissertation.

1.2 Context

Many underwater robots are available in the market or inside research laboratories.

An overview of such robots can be seen in [Yuh, 2000]. Underwater vehicles are designed

to suit specific applications and their development is in growth due to the high demand

in various fields where they are needed. They are capable of operating in environments

considered to be beyond the reach of divers. Moreover, they can be used in hazardous en-

vironments and can operate as long as needed 24 hours a day when tethered. In this thesis,

we are particularly interested in Remotely Operated Vehicles (ROV) for inspection applica-

tions. Inspection ROVs are small underwater vehicles dotted with a tether. Their weight

varies between 3 kg such as the AC-ROV (cf. Figure 1.1a) and 55 kg (including ballast for

sea water) for the Ocean Modules V8 Sii (cf. Figure 1.1c). These robots have various char-

acteristics in what concerns their size, weight, manoeuvrability, and embedded sensors.

To give an overview of common features of commercial inspection ROVs, figure 1.2 sum-

marizes the main specifications of 5 different ROVs. The specificity of each robot makes it

more suitable for a certain application rather than another. ROVs are used for many appli-

cations and some of them are listed here below:

1.2. CONTEXT 7

6.142.153.20 ×× 4.255.2453 ×× 265.4452 ××215.2235 ×× 507080 ××

3 6.3 4.10 13 60

75 76 150 300 1000

5 3 4 5 6

6 3 4 5 8

Figure 1.2: Comparative table among some commercial mini ROVs

Dam inspection

According to safety regulations, dams should be inspected every 10 years. This task is

nowadays undertaken by a robot controlled via a joystick by a certified pilot who receives

orders from the civil engineer in charge. The ROV is equipped with a camera and performs

a vertical scanning to inspect the state of the joints, and the state of the dam wall. Using

an underwater vehicle avoids the need of emptying the dam of its stored water which is

expensive given its energetic value. This mission lasts for over a week and can be imprecise.

To overcome these two drawbacks, some authors have proposed solutions for automated

inspection using a ROV [Maalouf et al., 2012a]. This not only improves the coverage rate of

the inspection, but also allows the mission to be performed by+ a less experienced pilot.

Ship hull inspection

The hull of boats have to be regularly checked for cracks, state of ware-markers or any-

thing unusual (mine, drug...). This is a difficult task for lengthy cruise ships or for offshore

vessels like FPSO (floating production, storage and offloading) (cf. Figure 1.4). An au-


Figure 1.3: An example of a trajectory for automated dam inspection by an underwatervehicle. Systemic scanning using constant intervals of depth.[Maalouf et al., 2012b]

tomatic inspection using a robot can avoid the need of dry docking the ship and hence

significantly reduce the inspection time. MIT and Bluefin robotics developped in [J. Va-

ganay, 2005] a hovering underwater vehicle conceived for missions concerned with anti-

terrorism and force protection. The implemented approach is easy to use by any opera-

tor and it is based on an inspection strategy having either horizontal or vertical slicing as

shown in Figure 1.5. The hull detection and the positioning of the ROV are achieved with

a DVL (Doppler Velocity Log). This latter is composed of 4 acoustic transducers. Distance

and orientation are measured using the 4 travelling times of the sound waves along the 4

beams, and the position is obtained by integrating the measured speed vectors.

Inspection of offshore structures

Offshore gas and oil exploitation comes along important subsea equipments and in-

stallations needing maintenance, inspection and repair. Underwater vehicles are not only

used to inspect pipelines, risers and windmills underwater foundations, but also to accom-

plish missions where manipulation is required (e.g valve manipulation). Oil industry can

be considered to be the most important end user of underwater robots. In Figure 1.6 is

depicted a marine drilling riser being the pipe linking the platform to the seabed.

Aquaculture

Today, underwater robotics is not only restricted to heavy duty applications but it also

1.2. CONTEXT 9

Figure 1.4: Total floating production storage and off loading (http://www.sjcho.com/)

Figure 1.5: Two approaches of hip hull inspection using horizontal or vertical slices [J. Va-ganay, 2005]

finds its place in the marine environment. Fish farming is highly affected by biofouling

which can increase the mortality of the fish due to the accumulation of micro-organisms

or algae under the cages or on the surface of the nets. Moreover, the nets can be dam-

aged and their holes have to be detected. For this reason, inspecting the nets is a regular

and necessary task to be undertaken. The usual methods for inspection and cleaning are

time consuming and expensive. In [Borovic et al., 2011] an ultrasonic underwater robotic

system is presented for this application. The system is easily deployable and operated (cf.

Figure 1.7).

Harbour installation structures

Inside a port, the inspection activities of a ROV are numerous. They can be useful for

the inspection of any kind of installation such as pontoons and docks. Some periodical

inspection should be carried out and they include the checking of electrical equipments


Figure 1.6: Marine drilling riser (http://oilandgastechnologies.wordpress.com/2012/08/27/steel-catenary-risers-scr/)

Figure 1.7: Underwater vehicle for cleaning of nets [Borovic et al., 2011]

and devices, and corrosion and ageing of harbour structures. Other than that inspection

regarding some safety measures related to plant facilities in the port can be undertaken.

1.3. ASSISTING THE PILOT 11

1.3 Assisting the pilot

Our aim in this thesis is to assist the pilot so that the ROV accomplishes its task partly

autonomously. In fact, the teleoperation of this type of vehicle is difficult since the exe-

cution of most of the tasks requires the simultaneous monitoring of various parameters

and degrees of freedom at the same time. The pilot often needs to use two joysticks while

proceeding very carefully in order to maintain a certain level of precision. Usually, the

robot’s operator is an expert who has followed several training sessions in order to acquire

the skill of piloting underwater vehicles. Having realized the complexity of teleoperation,

the manufacturers of ROVs have progressively improved their systems by equipping them

with additional features. The auto-depth option stabilizes the ROV at a designated depth.

The auto-altitude option stabilizes the ROV at a certain altitude from the seabed, and the

auto-heading fixes the robot on a specified magnetic heading. Some vehicles also have the

"freeze" option allowing them to be stabilized temporarily by maintaining the last orders

sent to the thrusters.

Automating the tethered vehicle will therefore facilitate various missions especially the

ones involving station keeping or systematic longitudinal scanning such as the inspection

of dams, boat hulls and pipes where the vehicle can be easily preprogrammed to follow a

prescribed trajectory. The aim behind the control is to determine the needed forces and

moments to be delivered from the actuators in order to accomplish the desired task. This

will require some feedback information from the available sensors to be fed into an algo-

rithm allowing the underwater vehicle to accomplish a set point regulation, path following

or trajectory tracking.

Different challenges in controlling such systems arise from the inherent high nonlin-

earities and the time varying behavior of the vehicle’s dynamics subject to hydrodynamic

effects and disturbances. The underwater environment is unstructured, non-uniform, and

varying. This adds complexity to the control of such systems since the dynamic model of

the robot cannot be fully determined given that some parameters are hard to compute

and are seldom constant (hydrodynamic coefficients, nonlinear damping ...). In fact, the

model parameters are likely to change with the environment and the mission. For example,

when the robot is required to manipulate objects, or carry payloads, or even be equipped

with additional sensors, its weight changes, as well as the centers of buoyancy and gravity.

Other common examples are the change of buoyancy when the water salinity varies, or the

damping increase when some algae gets a grip on the vehicle. Trajectory tracking involves

also accounting for some expected or unexpected external disturbances such as waves that

are common in shallow waters, or random obstacles that the vehicle might fail to avoid.

Since we are interested in tethered vehicles, it is important to mention that the umbili-


cal causes an important disturbing drag on the vehicle especially for smaller robots. It is

therefore desirable to design a controller able to deal with the inherent complex dynamics

of the system, while being robust to compensate parameter changes and overcome exter-

nal disturbances. Most of the control methods currently available on the commercial ROVs

rely on PD (Proportional Derivative) or PID (Proportional Integral Derivative) approaches.

The precision and the robustness of these methods is not high. In fact, the precision of the

depth regulation is often worse than 10 cm. Oscillations are often observed leading to a

degradation in the video quality, or to difficulties to catch objects with the manipulator.

The work that will be presented in the following chapters concerns the study of the

depth and pitch control of a commercial ROV (AC-ROV from the AC-CESS company). The

aim is to improve the stability and precision of the underwater vehicle in closed loop when

tracking a desired trajectory. The work involves a translational degree of freedom (the

depth) and a rotational one (the pitch) and it can thus be extended to the remaining ones.

The objective is to reduce the complexity of the operator’s work and improve the quality

of the ROV’s mission. Our study will therefore target the conception and application of an

advanced control scheme having a self tuning ability in order to maintain the performance

of the robot whenever changes occur in the dynamics or the environment.

1.4 Main contributions of the thesis

The main contribution of this thesis lies in the design, testing and full implementa-

tion of a novel controller in the field of underwater robotics. It is based on a recent con-

trol scheme that appeared in 2010 and was mainly tested on aerial vehicles. This thesis

presents an enhanced version of this controller in order to improve it in terms of trajectory

tracking. Experimental results were conducted on an underwater vehicle validating the

efficiency and robustness of the proposed solution. In particular, the thesis presents:

• Experimental results comparing conventional controllers, namely the PID controller

and the nonlinear adaptive state feedback controller. These controllers were tested

and compared in two degrees of freedom and in various scenarios in order to put

the vehicle in situations similar to the real ones in terms of parameter variation and

external disturbances.

• The adaptation, design, and application of the L1 adaptive controller which is a re-

cent controller. This controller has not made its entry in underwater robotics yet but

it is proven to decouple robustness from adaptation yielding higher performances.

• An improvement in the architecture of the L1 adaptive controller in order to provide

the system with a better closed loop performance in terms of trajectory following.

1.5. OUTLINE OF THE THESIS 13

A stability analysis has also been provided and experimental results validated the

efficiency of this new method.

• Real-time experimental comparison of the four above mentioned different control

schemes that have been tested and compared in two degrees of freedom and in vari-

ous scenarios on the same vehicle.

• Simulation results concerning roll stabilization with an internal disk with a detailed

description and calculation of all the dynamical effects of the thrusters on the dy-

namics of the vehicle.

1.5 Outline of the thesis

Chapter 2 presents the state of the art on the control schemes available in underwater

robotics. The methods provided represent an overview of what is mainly implemented

whether in simulation or in real-time experiments.

Chapter 3 addresses the vehicle dynamic modeling. This includes the frames used and

the equations of motions needed by the controllers for the establishment of the algorithms.

In addition to that, the full effects caused by the thrusters’ dynamics on the orientation of

the vehicle will be calculated.

Chapter 4 presents two conventional controllers in the field of underwater robotics.

The background on these controllers will be derived along with their application on depth

and pitch for an underwater vehicle.

Chapter 5 introduces a new controller in the field of underwater vehicles. It concerns

the L1 adaptive controller known for its robustness being decoupled from adaptation. A

description of the architecture and concept of this controller is given along with its design

and application on an underwater vehicle in depth and pitch.

Chapter 6 introduces an extended version of the controller presented in the previous

chapter. An augmented block will be added to the original architecture to achieve a better

performance in terms of trajectory tracking. The stability analysis will also be provided.

Chapter 7 presents the experimental platform used. A description of the test-bed will

be given along with the hardware architecture and the measurements of the needed state

variables for the feedback of the controllers.

Chapter 8 displays all the conducted experiments on the test-bed described in the

previous chapter. All control schemes have been implemented and compared among

themselves resulting in a performance study of the closed-loop system under various con-


trollers. The focus is the robustness towards unmodeled dynamics and the ability to reject

external disturbances while tracking a desired trajectory.

1.6 Conclusion

This chapter introduced the grounds on which this thesis is based. An overview of

the inspection applications and challenges that the human operator faces have been pre-

sented. We are interested in control schemes for trajectory following of a small ROV under

uncertainties, parameter changes and disturbances. The degrees of freedom to be con-

trolled are the depth and pitch and the objective is to achieve a better trajectory tracking.

The next chapter will present a state of the art concerning the available and already imple-

mented methods in this area.

CHAPTER

2State of the art on control schemes

Yesterday is but today’s memory,

and tomorrow is today’s dream.

KHALIL GIBRAN

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Classification of main existing control schemes . . . . . . . . . . . . . . . . 16

2.3 Classical control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Robust control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Adaptive control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Intelligent control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Hybrid control schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8 Comparison between the various schemes . . . . . . . . . . . . . . . . . . . 32

2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1 Introduction

Various challenges in automatic control arise when an underwater vehicle is used to

perform a mission as discussed in the previous chapter. Accomplishing such a task with-

out a human intervention means designing a control scheme able to deal with the highly

nonlinear behavior of the system along with the hostile operating environment and all the

15

16 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES

uncertainties of the model parameters. Taking the example of a dam inspection, the robot

needs to scan a large surface dragging its tether behind (if it is tethered) and rejecting the

disturbances coming from random obstacles encountered (rocks, algae, metallic rods pro-

truding from a wall etc ...). For this reason, precision and repeatability are required and

depending on the operating environment or load carried, some variations occur which can

destabilize the system. Taking all these criteria into account, various researchers and con-

trol theorists developed and implemented different methods with the aim of optimizing

the performances of the robot. In this chapter, we will discuss some of the techniques ap-

plied on underwater vehicles and validated in simulations or real-time experiments. The

list is not exhaustive but it gives a good overview about what is currently available in the

field.

2.2 Classification of main existing control schemes

Approximately from the year 1990 onwards, control methods have been proposed and

implemented both in simulations and real-time experiments for Unmanned Underwater

Vehicles (UUV). An overview of some of the related work can be found in [Yuh, 2000] and

[Budiyono, 2009].

The control strategies present today are numerous and different in theory and concep-

tion. For example, some linear methods are applied at each operating point. Usually such

methods are used when the UUV (Unmanned Underwater Vehicle) has no dominating

speed and can be linearized under several assumptions. Otherwise, for high performances

in different operating conditions, nonlinear modeling and control can be proposed. Non-

linear control may have the advantage of improving the robustness by taking into account

the nonlinearities present in the model or caused by the environment. This can be more

intuitive if the model is fairly precise since the physical properties of the system are taken

into account. Some techniques can be based (or not) on some a priori knowledge of the

system (weight, inertia, damping, etc ...). We classify such techniques as model-based or

non-model-based. On one hand, the methods that are model-based need to go through

the process of parameter identification. This can be a very cumbersome task especially

when it comes to evaluating the hydrodynamic coefficients. On the other hand, the non-

model-based controllers can be hard to tune requiring lots of trial and error testings before

getting the adequate gains.

Based on these ideas and on what is available in underwater robotics control, a classi-

fication of the main classes of control schemes is presented in the block diagram of Figure

2.1) with a focus on the following categories of schemes, namely:

2.2.C

LA

SSIFIC

ATIO

NO

FM

AIN

EX

ISTIN

GC

ON

TR

OL

SCH

EM

ES

17

Hybrid

SchemesIntelligent

Schemes

Adaptive

Schemes[Fossen et Fjellstad, 1996]

[Li et al., 2004]

[Sun et Chea, 2009]

[Zhao et Yuh, 2000]

[Antonelli, 2007]

[Fossen et Sagatun, 1991]

[Bessa et al., 2008]

[Marzbanrad et al., 2011]

[Zhou et al., 2010]

[Kim et Yuh, 2001]

Control Schemes

[Perrier et Canudas-De-Wit, 1996]

[McPhail et Pebody, 1997]

[Ostafichuk, 2004]

[Liu et Wang, 2005]

[Refsnes et al., 2005]

[Mirhosseini et al., 2011]

[Bian et al., 2010]

[Chang et al., 2003]

[Szymak et Malecki, 2008]

[Shi et al., 2007]

[El-Fakdi et Carreras, 2008]

[Lamas et al., 2009]

[Casalino et al,2012]

Classical

Schemes [Pan et Xin, 2012]

[Roche et al., 2011]

[Salgado-Jimenez et al.]

[Pisano et Usai, 2004]

[Campa et al., 1998]

[Le bars et Jaulin, 2012]

Robust

Schemes

Figure

2.1:Classifi

cation

ofth

em

ainco

ntro

lschem

esin

un

derw

aterro

bo

tics


– Classical schemes

– Robust schemes

– Adaptive schemes

– Intelligent schemes

– Hybrid schemes

2.3 Classical control schemes

Classical control schemes concern the commonly used methods encountered in lit-

erature. PID (Proportional Integral Derivative) control and its variants remain the most

widely used controllers. They can be easily implemented, are model independent and well

understood by everyone close to the control community. However, additional care should

be used with PID based schemes for underwater vehicles because the studied system is

highly nonlinear, varying, and coupled which might result in an unstable closed-loop be-

havior given the lack of robustness in this method. In addition to that, the tuning of the

controller’s feedback gains is not intuitive since it requires a knowledge of the system’s

characteristics and performance. This results in many trial and error testings on the field

before obtaining the adequate gains. Moreover any change in the experimental conditions

(e.g. additional payload, additional drag ...) requires retuning the controller. Other lin-

ear techniques consist in deriving a linearized model of the system around an equilibrium

point and then designing the controller based on the linear model. Some classical non-

linear control methods rely on the equations of motion and the dynamic model, such as

nonlinear feedback linearization. The problem in this case would be to identify the model

parameters. The performance of the controller is therefore highly dependant on how close

it is to the presumed known parameters of the model. Here below will be summarized

some references to those classical techniques applied in underwater vehicle control.

In [Perrier et Canudas-De-Wit, 1996] a nonlinear PID controller is proposed by adding

a nonlinear feedback loop to the classical PID scheme. The aim is to improve the sta-

bility and the disturbance rejection ability of the closed-loop system. The design of this

new method starts with the tuning of the traditional PID followed by the integration of

the nonlinear part which is summed to the PID input as shown in Figure 2.2. An experi-

mental comparison between the classical PID and the nonlinear one is performed on the

Vortex vehicle, a remotely operated vehicle of 150 Kg dry weight. Various scenarios were

implemented including heading, depth control and wall following. The results showed the

superiority of the proposed nonlinear extension in terms of fast response, disturbance re-

jection and overshoot cancellation in comparison with the classical PID. Figure 2.2 shows

a block diagram of the proposed control scheme. q refers to the states of the system,G(s)

2.3. CLASSICAL CONTROL SCHEMES 19

Figure 2.2: Block diagram of the PID controller proposed in [Perrier et Canudas-De-Wit,1996]

is the transfer function,H(s) a lead lag filter used to cancel the thruster low dynamics rep-

resented by BA andUNL the added nonlinear feedback.

In [McPhail et Pebody, 1997], the control and navigation systems of the autonomous

underwater vehicle Autosub-1 are described and tested. Experimental results are shown

for depth and pitch control using a PD controller. The design of the proposed method is

based on a cascaded control including two loops with the pitch control as the inner loop

and the depth as the outer one. Figure 2.3 shows a block diagram of the proposed control

algorithm. The displayed experimental results were carried outside Portland Harbour. The

mission required the vehicle to follow a squared reference trajectory at a depth of 10 m

before surfacing. The vehicle needed to surface and wait for 5minutes in order to acquire

GPS data. The times to first GPS fix varied between 27 to 42 seconds. During this mission,

the waves were of 2 m amplitude with 4 seconds period. The performance of the closed-

loop system was given in terms of the root mean squared values of the depth and pitch

(4 cm for the depth and 0.21 deg for the pitch).

Pitch

LimitDepth

Depth Demand

Max pitch

Min pitch

Pitch Rate

(Pitch Demand) Stern Plane Demand

Figure 2.3: Depth and pitch control algorithm for the AUV Autosub-1 [McPhail et Pebody,1997]


Navigation

ModuleController

Sensors and

Filters

Fuzzy Tuner

CompensatorSubmarine

Model

Noise

Disturbance

Figure 2.4: PD controller with fuzzy-tuned series compensation [Ostafichuk, 2004]

In [Ostafichuk, 2004] two variants of the classical PD controller are developed to im-

prove the control surface hydrodynamics for the Dolphin AUV. The proposed schemes re-

sult from the addition of two augmentations to the basic PD controller. In the first variant,

gain scheduling is used to change the coefficients during the operation and in the sec-

ond one, a fuzzy-tuned series compensation was added. The measured states are fed into

the fuzzy logic module that accounts for the changes in the control surfaces. The output

of this module serves to tune the parameters of the compensator as shown in Figure 2.4.

Numerical simulations showed that no significant difference in the performance of these

controllers was noted in terms of trajectory following and steady state error but a degrada-

tion was noticed when a parameter in the vehicle’s model was modified.

[Liu et Wang, 2005] designed a nonlinear output feedback controller for trajectory

tracking for the spherical AUV ODIN (Omni-Directional Intelligent Navigator). The dis-

turbances due to the waves when operating in shallow water were taken into account. An

observer has been designed to estimate this motion and the efficiency of the proposed

solution has been verified theoretically through the proof of stability and numerical simu-

lations.

In [Refsnes et al., 2005] an output feedback controller was implemented. Using the

dynamic model of the Minesniper MKII a torpedo shaped ROV, the controller was de-

signed considering current disturbances. The estimation of the current velocity is provided

through an observer which improves the tracking performance. In addition to that, this

work proposed an elaboration on the modeling of the hydrodynamic, coriolis forces and

moments that might destabilize the system when the vehicle moves at a relatively high

forward speeds. Numerical simulations validated the proposed method.

2.4. ROBUST CONTROL SCHEMES 21

[Mirhosseini et al., 2011] use nonlinear control theory in output regulation for seabed

tracking for an AUV using the model of the Medusa [Gantenbrink et Victor, 1983], an AUV

weighing 140 kg. The sea bottom is considered sinusoidal but the vehicle is not aware of its

profile in advance. It was shown through simulations that the proposed method is capable

of maintaining the vehicle at an offset constant distance from the seabed using a single

echo sounder sensor.

[Bian et al., 2010] design a nonlinear controller based on the input-state linearization. A

longitudinal underwater vehicle is considered and the objective is to perform a trajectory

tracking in the horizontal plane. Simulations have been performed by taking the rudder

angle as the control input and the position in the horizontal plane as the controlled output.

The technique of pole placement was used to design a virtual input for trajectory tracking.

The resultant system is therefore a linear one transformed as such through state and in-

put transformation with state feedback. The proposed control scheme was compared to

a classical PID controller. Simulation results show that the performance of the nonlinear

controller is better in terms of trajectory following and external disturbance rejection.

It can be concluded from the listing of these control schemes that the main concern

of such methods is to achieve a desired tracking. The PID based control techniques [Per-

rier et Canudas-De-Wit, 1996][McPhail et Pebody, 1997][Ostafichuk, 2004] require an ad-

equate tuning and are usually able to follow the reference trajectory. However, when it

comes to robustness towards parametric uncertainties and disturbance rejection, a clear

degradation of the system’s performance in closed-loop is observed. The classical nonlin-

ear control schemes such as output feedback/regulation or input state linearization ([Liu

et Wang, 2005][Refsnes et al., 2005][Mirhosseini et al., 2011][Bian et al., 2010]) have mostly

been only tested in simulation. Given the fact that they take the nonlinear dynamics of the

system into account, these methods can be more advantageous than the PID based ones if

they have in disposal a precise dynamic model. Indeed, they have a better ability to reject

external disturbances, however no robustness to parameter change can be guaranteed. In

summary, these methods are easy to implement and are often designed for trajectory fol-

lowing but they lack robustness to parameter change as well as a disturbance rejection

ability.

2.4 Robust control schemes

The control of underwater vehicles is a challenging task as seen previously and even if

classical control schemes brought fairly good results, the high nonlinearity and coupling in

the dynamics of the system, the environmental disturbances and the uncertainties, min-


imize the efficiency of these methods. A controller with a robustness ability is capable

of maintaining the stability of the system despite the variations in the model (e.g. pay-

load changes, evolution of thrusters’ performances, etc) and the operating conditions (e.g.

salinity, mechanical impacts, tether, drag, etc). In addition to that, such schemes are also

designed to guarantee a desired closed-loop performance in terms of steady state error,

convergence time etc. One of the common robust methods is high gain feedback where the

effects of the variations in the model are made negligible through the large imposed gain.

The controller is therefore static, it can maintain the stability of the system while assum-

ing that some parameters will remain unknown. The two most common robust methods

in underwater robotics are the H∞ and the sliding mode control. The former method is

linear whereas the latter one can be composed of a linear part and a nonlinear part or be

entirely nonlinear.

[Pan et Xin, 2012] propose an indirect robust controller for depth regulation for the

REMSUS AUV. The robust scheme is considered indirect because the uncertainty bounds

are formulated into a cost function to be optimized. The control problem is therefore trans-

formed into an optimization problem. Solving the optimal problem will lead to finding the

necessary feedback control law guaranteeing robust asymptotic stability. Simulation re-

sults show that the performance of the robot is conserved when white noise is added to

the model parameters.

Depth

controller

Pitch

controllerSubmarine

Model

Figure 2.5: Cascade control configuration for altitude control [Roche et al., 2011]

[Roche et al., 2011] propose a cascaded architecture including two controllers for depth

and pitch using the H∞ framework when the AUV is subject to real-time constrains. The

depth controller sets a pitch reference to the pitch controller that will compute the neces-

sary control input (cf. Figure 2.5). This architecture was possible because the motions in

the vertical plane and around the pitch axis are generated by different actuators. A linear

parameter varying model of theAsterXAUV was built by taking into account the sampling

2.4. ROBUST CONTROL SCHEMES 23

time as the varying parameter. The objective is to render the controller robust in presence

of asynchronous measurements in the control algorithm. In simulations, the nonlinear

model of the vehicle on which the controller was applied was considered. Results show

that stability has been preserved when variations occur on the sampling time.

[Salgado-Jimenez et al., 2004] compare in simulations the behavior of an AUV in closed-

loop under a first order sliding mode and a high order sliding mode controllers. Sliding

Mode Control (SMC) is a robust controller designed to deal with strong uncertainties by

reacting immediately to any deviation of the system stirring it back to the constraint by a

sufficient energetic effort. It is the derivative of this deviation that differs between these

two sliding mode controllers. In fact, a higher order time derivative of the deviation is

used for the higher order SMC. This results in eliminating the chattering effect (very high

frequency oscillations occuring in the actuators action) by smoothing the control input.

Sliding mode controllers based on twisting and super twisting algorithms are used. The

term twisting refers to the trajectory of the sliding variable on the manifold. This variable

follows an infinite number of decreased rotations to converge to the origin. The simulation

results showed that the higher order SMC decreased the chattering effect and improved the

tracking precision.

In [Pisano et Usai, 2004] a jet propelled underwater vehicle is controlled in closed-loop

with an output feedback controller by a second order SMC. A second order sliding mode

differentiator is also used in order to provide a more adequate estimate of the derivatives of

the tracking error in presence of a measurement noise. The prototype is connected with a

wheeled trolley and moves along a water channel. The variable to be controlled is the posi-

tion of the vehicle in this channel. Two experimental tests have been carried out in order to

test the proposed method. The first one involved a piecewise constant reference position

and the second one was a tracking test with a sinusoidal reference trajectory. A compar-

ison with a classical PID controller for constant reference trajectory shows that the SMC

is more accurate despite the fact that the convergence is slower compared to the PID con-

troller. Another test with the proposed method was carried out using a sinusoidal reference

trajectory and the results showed an adequate tracking with a short transient convergence

time.

[Campa et al., 1998] evaluate the robustness of two controllers implemented in simu-

lation on a missile-like AUV for attitude and position control around an operating point

of interest. The first scheme is the µ technique which is a linear approach and the second

one is the SMC technique. The µ synthesis is based on the framework shown in Figure

2.6 where for unstructured systems all sources of uncertainties occur at a specific location

in the loop (Delta in this case). The control objective is to find the controller K in order to


Plant

Output

Disturbance

Delta

K

Input

Error

Figure 2.6: Framework of a µ analysis based robust scheme [Campa et al., 1998]

minimize the structured singular value of the plant’s transfer function which refers to look-

ing at the matrix Delta and reducing the system’s sensitivity towards its effect. Simulation

results show that the SMC outperformsµ in presence of high inherent nonlinearities but is

less efficient in a narrow range around the operating point.

[Bars et Jaulin, 2012] present a robust controller for the sailboat VAIMOS. This surface

robot has been designed for oceanographic purposes. In fact it measures ocean parame-

ters near the water surface and it has several sensors (GPS, wind sensor, compass) and a

WIFI and Iridium communication system. In order to cover an area as accurately and au-

tonomously as possible, a robust line following controller was designed and implemented.

The efficiency of the method was validated theoretically using interval analysis and Lya-

punov methods and later on using a simulator with the hardware in the loop. This step

was a preparation for the real-time experiments in the ocean where several missions were

made. Two of them near the harbor of Brest (for a distance of 8 km and 14 km). The sail-

boat had to deal with different wind conditions and trajectories. A mission over more than

100 km was made between Brest and Douarnenez. The robustness of the controller was

shown through the perturbations caused by the presence of obstacles in the way. Despite

the forced deviations, the sailboat was able to continue and reach its destination.

Robust control schemes can deal with strong uncertainties present in the system or the

operating environment while preserving the stability of the closed-loop system. Design-

ing such a control scheme, releases the need of having a precise model for the vehicle. In

fact, a good estimation of the model’s parameters is not needed, but it is required to have

a boundedness on the uncertainties. Disturbance rejection is also guaranteed and this al-

lows the robot to perform its designated tasks in unknown operating conditions and in

presence of disturbances [Bars et Jaulin, 2012]. Often for such methods like sliding mode

control or high gain feedback, the control input is either discontinuous or holds high fre-

quency oscillations which can deteriorate the actuators [Salgado-Jimenez et al., 2004]. The

2.5. ADAPTIVE CONTROL SCHEMES 25

Plant

Parameter

Estimator

Controller

Figure 2.7: Direct adaptive control method

performances of a robust controller can be limited if large uncertainties are present. For

this reason, better results can be obtained when a coupling with an adaptive scheme is

done. In fact, these two methods handle parameter variations and uncertainties despite

the differences they hold. Adaptive control does not need a known uncertainty bound but

rather adapts the controller, whereas robust control guarantees a good performance with

the same controller within the given bounds.

2.5 Adaptive control schemes

With the advances of robotics and the industrial growth, various challenges in non-

linear control saw the light. Among the popular schemes dealing with varying systems

and robustness, adaptive control was born. Such a controller has the ability to adapt to a

system having varying or unknown parameters. In fact, these schemes evaluate the per-

formance of the closed-loop controller and retune it autonomously. Adaptive techniques

span into two main categories: direct methods and indirect methods. For the direct meth-

ods, the control parameters are estimated directly as seen in Figure 2.7 and then the plant

model is readjusted accordingly. As for the indirect methods, the plant parameters are the

ones estimated and then used to design the controller. Figure 2.8 shows a sketch of this

method. From what has been seen so far, we can therefore deduce that these schemes are

considered to be dynamic compared to the robust ones where we consider some parame-

ters remain unknown under some assumptions. In underwater robotics, adaptive control

has been widely used given the above advantages it offers. Many parameters can change


Plant

Parameter

Estimator

Controller

Controller

Design

Plant Parameters

Figure 2.8: Indirect adaptive control method

during the mission of the vehicle and should be updated. For example, the addition of

sensors or payloads modifies the weight and the position of the center of mass of the ve-

hicle. Furthermore, the experimental conditions vary whether the vehicle operates in the

sea or in the river resulting in a different buoyancy for each case. Various control methods

were elaborated under the family of adaptive control. The dynamics of most systems is

linear with respect to the model parameters and therefore it can be written as a regressor

matrix multiplied by a vector of parameters. This latter is updated at every iteration us-

ing the adaptation law combining the tracking error, the regressor and an adaptation gain.

This gain usually plays the role of a trade off between fast adaptation and robustness. Here

below is a list of the main adaptive control schemes applied on underwater vehicles.

[Fossen et Fjellstad, 1996] present a comparative study between the two adaptive con-

trollers [Slotine et Benedetto, 1990] and [Sadegh et Horowitz, 1990] in terms of robustness

towards measurement noise. In [Slotine et Benedetto, 1990], the regressor and the desired

torques are computed using the measured state values of the system whereas in [Sadegh

et Horowitz, 1990] they are computed using the desired ones. For this reason, this latter

controller is also called the "desired compensation law". Simulation results performed in

presence of unknown model parameters and noisy measurements showed that a better

performance was obtained using the method proposed by [Sadegh et Horowitz, 1990].

[Li et al., 2004] present an adaptive controller for the diving motion of an AUV. The

specificity of the proposed scheme lies in the formulation of the problem where some clas-

sical assumptions are broken. The pitch angle is not considered small and the pitch mo-

tion dynamics is not expressed as a linear equation. Considering these assumptions may

2.5. ADAPTIVE CONTROL SCHEMES 27

induce large modeling errors causing severe problems in practical applications. The con-

troller is designed using the backstepping technique. Numerical simulations have been

made to show the efficiency of the method. To avoid the divergence of the estimated pa-

rameter vector, a parameter excitation is needed. Given that this excitation is hard to be

achieved, the adaptation law needed to be modified and it included design parameters and

coefficients calculated for the tested underwater vehicle.

[Sun et Chea, 2009] proposed two adaptive proportional-derivative control laws. Both

schemes require only the gravity regressor instead of the full one for the whole dynamic

model. The transformation matrix transpose was used instead of the inverse for the map-

ping between body and earth frame. The first controller is an adaptive setpoint controller

and the second one is a region reaching controller also considered to be is a generalization

of the setpoint control. The stability analysis was provided through lyapunov like func-

tions. Numerical simulations were performed on the model of the omni directional vehicle

ODIN. For the adaptive setpoint control, a desired position for the six degrees of freedom

was specified, and the vehicle was able to converge within 5 s. Similarly for the region

reaching control, the vehicle was required to reach a desired area defined by individual re-

gional bounds. In the presented simulations, the region was specified by a parallelepiped.

The time needed for the error to converge to zero depends on how far the initial error is

but it is always less than 10 s.

[Zhao et Yuh, 2000] present a nonregressor based controller to avoid the need of having

some knowledge of the dynamic model and estimate a large set of parameters. The scheme

proposed by the authors also includes a disturbance observer. Interesting experimental re-

sults have been obtained in nominal conditions as well as robustness towards parameters’

uncertainties and external disturbances rejection. The advantage of this method is that it

does not require a priori knowledge of the system; furthermore the update of the param-

eters is based on the performances of the closed-loop system. However, the drawbacks

of such a method include the neglect of the coupling effects between the degrees of free-

dom since the validation was performed on a spherical vehicle. The model parameters of

the dynamic model can be initialized randomly, but the control parameters governing this

method are very critical to be chosen and highly dependent on the initial configuration of

the robot.

[Antonelli, 2007] compares the following six adaptive controllers: [Fjellstad et Fossen,

1994a][Yuh et Nie, 2000][Sun et Chea, 2003][Fossen et Balchen, 1991][Fjellstad et Fossen,

1994b] and [Antonelli et al., 2001] in simulation within a study that focuses on their ability

to compensate for the persistent effects (restoring forces and ocean currents). The non-

regressor-based methods [Yuh et Nie, 2000][Sun et Chea, 2003] were unable to compensate


for the restoring forces and the model-based methods [Fjellstad et Fossen, 1994a][Fossen

et Balchen, 1991][Fjellstad et Fossen, 1994b] needed adequate persistent excitation. This

will generate a problem at steady state when a static error occurs in presence of waves or

current. In fact, in this scenario, the parameter excitation will be reduced since the error on

the velocity is zero while the position error is not, and therefore a corrective adaptive con-

trol action cannot be triggered. The adaptive control law introduced in [Antonelli et al.,

2001] was the one defended in this comparative paper because it accomplishes the de-

sired full compensation. However, it still requires the adaptation of nine parameters with

a suitable initialization of the restoring parameter vector and a reasonable choice for the

adaptation gain. The simulations were performed on an ellipsoidal autonomous vehicle

weighing 225kg.

Adaptive schemes provide the controlled system with a self-tuning ability. An online

adaptation takes place if uncertainties are present in the model parameters and if these

parameters change during the vehicle’s mission. Some adaptive controllers are regressor

based as seen above [Fossen et Fjellstad, 1996] and all of them need the adaptation of a

set of parameters. This adaptation is made easier when an excitation is performed and

also when the parameter initialization is performed based on some a priori knowledge of

the controlled model [Antonelli, 2007]. The convergence of the closed-loop system to the

desired trajectory depends on the time needed for the parameters to converge and this is

closely related to the adaptation gain. The higher it is, the faster the convergence can be.

However, this can be at the price of deteriorating the system’s robustness and the transient

behavior causing thruster saturation or even instability.

2.6 Intelligent control schemes

Intelligent control includes the recent schemes elaborated with the scope of imitating

some specificities of the human intelligence. It bases its methods on biological systems

and for this reason it branches into different techniques. Among these methods we find

neural networks, fuzzy logic control, genetic algorithms, evolutionary algorithms, etc. The

control schemes usually require skills in artificial intelligence and computer science. Ar-

tifical neural networks for example were inspired by the biological central nervous system.

They are constituted of a set of adaptive weights undergoing a learning algorithm and ca-

pable of approximating nonlinear functions. Fuzzy logic is a reasoning based on approx-

imations instead of exact and precise input information. Evolutionary algorithms are a

wide class including for instance the genetic algorithms. These algorithms are based on an

idea similar to the biological evolution and the natural selection where an optimization is

made so that only the best solution remains.

2.6. INTELLIGENT CONTROL SCHEMES 29

[Chang et al., 2003] implement a Takagi-Sugeno (T-S) type fuzzy model on an under-

water vehicle. This fuzzy controller is model based and it uses the concept of Parallel

Distributed Compensation (PDC). The proposed scheme is described by IF-THEN rules

representing local input-output relations of the nonlinear system. The PDC concept al-

lows to construct the feedback gains for each rule. Numerical simulations were performed

on an AUV nonlinear model. Firstly, an open-loop test was performed on both the AUV

real dynamic model and the T-S fuzzy one showing that their behavior coincides. Tests in

closed-loop using the fuzzy controller were made for both systems showing a fast conver-

gence for the desired regulation in the tested degrees of freedom.

[Szymak et Malecki, 2008] propose a PD controller based on fuzzy logic for the under-

water remotely operated vehicle Ukwial. Computer simulations where initially performed

in the vertical and horizontal plane in presence of different sea currents. Experimental re-

sults in heading and depth were carried out in a calm sea and compared to the simulated

ones. Similarities were observed according to these results and it was concluded that the

tether can have a stabilizing effect for the heading and that its model is not reliable. For the

depth, the noise in the measurements worsened the experimental results.

[El-Fakdi et Carreras, 2008] present a high-level reinforcement learning scheme for the

autonomous underwater vehicle ICTINEUAUV being assigned the task of tracking a cable.

In order to reduce the time of the learning process, the artificial designed neural network

was trained in simulations prior to the real tests. The obtained results were later on trans-

ferred to the experimental setup which uses the same controller and the presented algo-

rithm was therefore validated.

[Lamas et al., 2009] develop and test an evolutionary algorithm with artificial neural

networks for the control of a submersible catamaran meant for tourists. For security and

comfort reasons, the control algorithm has some constraints and limitations regarding the

orientation angles and the accelerations. A hydrodynamic simulator was developed in or-

der to validate the proposed scheme that was implemented on the modeled system in pres-

ence of external disturbances.

Recently [Casalino et al., 2012] present a task and subsystem priority based control

strategy for an Intervention AUV (I-AUV). An I-AUV is an underwater vehicle dotted with

a robotic arm allowing it to perform activities underwater such as collecting objects. The

scope of the proposed algorithm is to guarantee the floating manipulation capabilities of

the robot. When the vehicle detects a target, it switches to the manipulation mode in or-

der to execute the required activity. For the completion of the mission, a set of inequality

and equality control objectives are to be achieved. The objectives to be fulfilled are the

joint limits, manipulability, the horizontal attitude and the position of the camera. Pe-


nality functions are used to assign priorities for the designated tasks. Successful results

were shown in simulations. They displayed the behavior of the I-AUV when needing to ap-

proach an object underwater and then center the camera according to its position while

respecting the task priorities and the joint limits.

To summarize, intelligent control methods imitate some biological systems and use ar-

tificial intelligence or algorithms inspired from human intelligence and biological systems.

Such schemes require skills in computer sciences and mathematics. They are often hard

to implement and need more computational time than other techniques. In addition to

that, some of them also need some thorough tuning [Shi et al., 2007], training [El-Fakdi et

Carreras, 2008] or trial and error cycles [Szymak et Malecki, 2008].

2.7 Hybrid control schemes

Various control schemes have been applied successfully on underwater vehicles. For

this reason, it can be highly beneficial to combine different techniques in order to marry

the advantages present in each one of them. For example, having a method putting to-

gether sliding mode control and adaptive control yields a robust controller with a self tun-

ing ability. Hybrid schemes are also useful when some known drawbacks are overcome by

the addition of a corrective action found in another technique.

[Fossen et Sagatun, 1991] present a hybrid controller combining adaptive and sliding

mode control. This work is considered one of the pioneers in proposing such schemes.

It consists of an online parameter estimator and a switching term compensating for the

uncertainties in the input matrix. The latter ensures the mapping between the desired

torque and the motors’ inputs. The objective is therefore to compensate for the uncertain-

ties while taking into account the time-varying behavior of this matrix that is caused by the

thruster hydrodynamics. Simulation results on the ROV NEROV were presented to show

the efficiency of the proposed method. The vehicle needed to follow a desired trajectory

in the horizontal motion (surge, sway and yaw). All tracking errors converged to zero in

around 15 s.

[Bessa et al., 2008] combine a sliding mode controller with an adaptive fuzzy algorithm

in order to enhance the compensation of the uncertainties and disturbances. In fact, to

overcome the chattering problem, many SMC (including this work) use a saturation func-

tion instead of the sign function at the risk of degrading slightly the tracking performance.

The adaptive fuzzy strategy proposed in this paper was designed to eliminate this draw-

back while preserving the closed-loop stability. Simulation results for depth control show

2.7. HYBRID CONTROL SCHEMES 31

the improvement brought by this combination over the conventional sliding mode con-

troller.

[Marzbanrad et al., 2011] validates in simulation a scheme similar to the previously

cited one [Bessa et al., 2008]. The main difference with this algorithm is that it has an added

robustifying control term. The scheme is therefore considered to be a robust adaptive

fuzzy sliding mode control. Its objective is to estimate online the external disturbances and

the unmodeled dynamics while guaranteeing a tracking error withing satisfactory bounds.

Successful simulation results were performed on the Ariana-I ROV.

[Zhou et al., 2010] propose a state feedback sliding mode controller to eliminate the

chattering phenomenon seen with the traditional SMC and without the need to refer to

higher order sliding mode schemes. The idea behind this proposed method is to combine

the advantages of two control methods. SMC can deal with the nonlinearities, uncertain-

ties and disturbances, while the state feedback controller improves the performance. For

this reason, the former controller deals with the nonlinear part of the system and the latter

with the linear one. The chattering phenomenon is eliminated through the eigen values

of the Hurwitz matrix that is built via the feedback imposed. The proposed controller was

compared to the classical SMC and the improved performance of the hybrid controller was

validated in terms of stability, and chattering elimination.

[Kim et Yuh, 2001] designed a controller with a self tuning ability by combining fuzzy

logic and neural networks. The proposed control scheme can therefore benefit from the

advantages of both algorithms. A human operator expertise is used in the definition of

the fuzzy rules and values whereas the learning ability of the controller is provided by the

neural networks. Simulation results performed on the underwater vehicle ODIN validated

the performance of the proposed method.

[Shi et al., 2007] developped a neural network based adaptive scheme to control the

depth of an underwater vehicle. The method serves to estimate the nonlinear parameters

using a feedforward neural network in order to track a desired depth. Simulation results

show that a satisfactory tracking was obtained but a better performance could be achieved

by further tuning the update gain or increasing the number of neurons in the architecture

of the scheme.

Hybrid schemes are designed in order to combine the advantages of different methods

yielding a better closed-loop behavior of the underwater vehicle. The objective is not only

trajectory tracking but also robustness, adaptation and stability [Bessa et al., 2008]. The

implementation of such control schemes require a precise knowledge of the chosen meth-

ods in order to adequately associate them. In case there is a switching term, a special care


should be made in order to avoid a discontinuity [Fossen et Sagatun, 1991]. Despite the

fact that such methods seem appealing, very few experimental validations can be found in

the literature.

2.8 Comparison between the various schemes

In this chapter, the main control schemes encountered in the field of underwater

robotics were presented. They have been classified in different categories according to

some common specifications they share. The list is not exhaustive, other methods were

also tested in simulations and experiments such as linear quadratic gaussian control

[W. Naeem et Ahmad, 2003] or predictive control [Steenson et al., 2012]. Comparing and

evaluating these methods is interesting in order to understand the strengths and weak-

nesses of each one of them. Comparisons between various controllers can be found in the

literature and mainly through simulations. For instance, in [Fossen et Fjellstad, 1996] and

[Antonelli, 2007] a comparison among adaptive controllers is reported. The former study

evaluates robustness of each control law against measurements noise and parameters’ un-

certainties while the latter describes the ability of each adaptive controller to compensate

for the currents and restoring forces. A sliding mode controller was compared in simula-

tions with the mu synthesis in [Campa et al., 1998] while a robust adaptive fuzzy sliding

mode controller in [Marzbanrad et al., 2011], in terms of trajectory following and mea-

surement noise. In [Smallwood et Whitcomb, 2002], four various model based controllers

(adaptive and nonadaptive exact linearizing controllers, adaptive and nonadaptive nonlin-

ear controllers) were experimentally compared with a PD controller in the case of a good

and a bad initial parameter estimation and in the case of thruster saturation.

The previously stated schemes reveal some strengths and weaknesses as brought up

through the work of various authors. The different categories listed above were compared

in Table 2.1 according to some relevant selected criteria. Each category has been voted by

a plus sign (+) ranging from 1 being the least favorable mark for the selected criterion to 5

being the most favorable one. We can deduce from the table that the robust, adaptive and

hybrid schemes have similar performances and seem to be convenient for our application.

2.9 Conclusion

The main categories of control schemes proposed in the literature of underwater

robotics have been presented in this chapter. Their strengths and drawbacks have been

highlighted through examples given by the work of various authors and by the summary

2.9. CONCLUSION 33

Table 2.1: Comparison among the di!erent schemes according to

some selected criteria

illustrated in the comparative Table 2.1. PID based methods are hard to tune and fail in

presence of parameter variations. Robust methods have a limited performance in pres-

ence of high uncertainties and adaptive schemes have their robustness characteristics and

convergence time tied to the chosen value of their adaptation gain. The intelligent meth-

ods based on neural networks require time for the training of the network or can be hard

to tune. Hybrid schemes can be interesting but no generalization can be made since they

depend on which controllers are combined and how the switching between them is per-

formed. Based on these arguments, we state the need for designing new controllers yield-

ing the desired closed-loop behavior of the underwater vehicle. According to the context

of study, the desired controller must have a self tuning ability while preserving the robust-

ness of the closed-loop controlled system. A fast convergence to the desired trajectory with

a satisfactory transient phase is sought despite the possible uncertainties, the changes in

the model and the encountered external disturbances.

CHAPTER

3Modeling of underwater vehicles

Luckily, I’m doing other things

besides just modeling, because

frankly, I’m a little bored with it.

REBECCA ROMIJN

Contents

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Thruster dynamic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1 Introduction

Modeling is a necessity whenever control comes into picture. A model is needed for

the design of various control schemes. In addition to that, the most important use of hav-

ing a model remains for simulation purposes. Control schemes are mostly always tested

in simulation first due to time and cost constraints. Having a model of the system to be

controlled helps in determining the behavior of the plant and gives an insight into the way

of formulating or designing the control algorithm. In this chapter, the vehicle kinematics

and dynamics will be described. The model of the vehicle will be derived according to the

35

36 CHAPTER 3. MODELING OF UNDERWATER VEHICLES

formulation proposed by Fossen [Fossen, 2002] using the standard Society of Naval Archi-

tects and Marine Engineers (SNAME) notation (1950). It is important to note though that

different other types of modeling for underwater vehicles exist in literature as reported by

[Newman, 1977],[Yuh, 1990], and [Leonard, 1996]. The purpose is to express through a

mathematical model the behavior of the underwater vehicle. To reach this objective, the

kinematics and the dynamics of the vehicle are to explained here below.

3.2 Kinematics

Kinematics is the branch of mechanics concerned with the motion of a body without

considering its mass or the forces acting on it. In summary, it deals with the position of

a system, the generation of trajectories and also the velocities and accelerations. For this

reason, the geometric aspects enter in the description of the vehicle and therefore the po-

sitions and velocities are defined. As mentioned before, the SNAME notation will be used

and it is illustrated in Table 3.1. The position of the vehicle is expressed via a set of six

variables representing six degrees of freedom (DOF). These DOF refer to three translations

and three rotations. They are expressed in two reference frames: the body-fixed frame and

the earth-fixed frame. The earth-fixed frame also called the inertial frame as illustrated in

Figure 3.1 is denoted as NED referring to a North East Down representation given that the

positive direction of z is downwards. The body-fixed frame is the one fixed to the vehi-

cle. Its position and orientation are described relative to the inertial reference frame. The

origin of this frame coincides with the center of buoyancy of the underwater vehicle. The

linear and angular velocities pertain to the body-fixed frame whereas the Euler angles and

the positions are absolute and pertain to the earth-fixed frame. However, the transfer from

one frame to the other is possible and it is achieved using the transformation matrix J. This

latter is a function of the euler angles and it is constituted of a translation matrix and a ro-

tation matrix. The euler angle representation is sequence of three rotations: roll, pitch and

yaw around the following the x, y and z axes respectively as shown in Figure 3.1. The vari-

ables being translated from a frame to another and expressing the position and velocity of

the vehicle in the two frames are summarized in Table 3.1 and can be represented by the

following vectors:

η= [x,y,z,ϕ,ϑ,ψ]T being the vector of position in the earth frame

ν= [u,v,w,p,q,r]T being the vector of velocities in the body-fixed frame

Ω= [p,q,r]T being the vector of angular velocities in the body-fixed frame

Θ= [ϕ,ϑ,ψ]T being the euler angles in the earth-fixed frame

(3.1)

3.2. KINEMATICS 37

Table 3.1: The SNAME notation for marine vessels

Motion Linear and angular velocities Positions and Euler angles

Surge u x

Sway v y

Heave w z

Roll p ϕ

Pitch q ϑ

Yaw r ψ

pitch

(sway)

(heave)

(surge)

rollpitch

yaw

Figure 3.1: View of an underwater vehicle and its reference frames (xiyizi: earth-fixedframe, xbybzb: body-fixed frame).

J is the transformation matrix ensuring the relationship between the vectors ν and η

according to the following:

η= J(Θ)ν (3.2)


J(Θ) ∈ R6×6 includes a rotation matrix R(Θ) and a translation matrix TΘ(Θ) and it is

given by:

J=

[R(Θ) 03×3

03×3 TΘ(Θ)

](3.3)

R(Θ) is expressed as follows:

R(Θ)=

cψcϑ −sψcϕ+cψsϑsϕ sψsϕ+cψcϕsϑ

sψcϑ cψcϕ+sϕsϑsψ −cψsϕ+sϑsψcϕ

−sϑ cϑsϕ cϑcϕ

(3.4)

and TΘ(Θ) is given by:

TΘ(Θ) =

1 sϕtϑ cϕtϑ

0 cϕ −sϕ

0 sϕcθ

cϕcϑ

(3.5)

In both formulas (3.4) and (3.5), the notation used follows this model cθ = cosθ,

sθ = sinθ and tθ = tanθ. It is also important to note that TΘ(Θ) crosses a singularity

point at ϑ=±π2 .

3.3 Dynamics

The underwater vehicle is a body with six degrees of freedom with a dynamics repre-

sented using two reference frames as seen above. This dynamics described here below

exposes the relationship between the torques applied on the system and the resulting mo-

tion generated. The system is considered highly nonlinear given the hydrodynamic effects

on the vehicle that are hard to model. By considering the inertial generalized forces, the

hydrodynamic effects, the gravity, and buoyancy contributions, the dynamic model of an

underwater vehicle in matrix form, using the SNAME notation and the representation de-

scribed in [Fossen, 2002], can be written as follows:

Mν+C(ν)ν+D(ν)ν+g(η)= τ+wd (3.6)

The model matrices M, C, and D denote inertia (including added mass), Coriolis-

centripetal (including added mass), and damping respectively, while g is a vector of grav-

itational/buoyancy forces. τ is the vector of control inputs and wd the vector of external

3.3. DYNAMICS 39

disturbances. In the case of our study, the vehicle used has a slow dynamics, and hence it

will be moving at velocities low enough to make the Coriolis terms negligible (C(ν)ν≃ 0).

Considering this assumption, the dynamics (3.6) can be simplified to:

Mν+n(ν,η)= τ+wd (3.7)

where n(ν,η)=D(ν)ν+g(η).

Equation (3.7) describes the dynamics of the system in 6 degrees of freedom taking into

account the 3 translations and the 3 rotations. The input vector τ ∈ R6×1 considers six

actions on the system to fully control it. The presented formulation of the robot’s dynamics

is expressed in the body-fixed frame and can be transformed to the earth-fixed frame using

the kinematic transformations of the state variables and the model parameters according

to the following:

η= J(η)ν

η= J(η)ν+ J(η)ν

M∗(η)= J−T (η)MJ−1(η)

D∗(ν,η)= J−T (η)D(ν)J−1(η)

g∗(η)= J−T (η)g(η)

τ∗ = J−T (η)τ

w∗d= J

−T (η)wd

(3.8)

with J being the transformation matrix described in the previous section. Equation (3.7)

can therefore be expressed in the earth fixed frame as:

M∗(η)η+D∗(ν,η)η+g∗(η) = τ∗+w∗d (3.9)

For more details about the model parameters, the reader is referred to Appendix E for a

complete description of the vehicle under study.

In this thesis, the control objective concerns two degrees of freedom: depth and pitch. For

this reason, we establish from the above model a reduced one that we express in the body-

fixed frame according to the following:

Mrν+Dr(ν)ν+gr(η)= τr+wdr (3.10)

with the subscript r denoting the reduced model. When writing explicitly the above

expression, we get:[Mz 0

0 Mϑ

][w

q

]+

[Dz 0

0 Dϑ

][w

q

]+

[cos(ϕ).cos(ϑ).(B−W)

WzGsin(ϑ)

]=

[τz+wdzτϑ+wdϑ

]

(3.11)


Mz and Mϑ are the mass parameters in the body-fixed frame pertaining to z and ϑ re-

spectively. Dz and Dϑ are the damping parameters in the body-fixed frame for these two

degrees of freedom as well. W is the weight of the vehicle, and B its buoyancy. zG is the z

coordinate of the center of gravity. Finally on the right hand side of the equation, we have

the control inputs summed to the disturbances expressed in the body-fixed frame for the

depth and the pitch.

Since the expression of the complete model was also expressed in the earth-fixed frame, the

reduced one will be expressed similarly as well. The general matrix form is the following:

M∗r (η)η+D

∗r (ν,η)η+g

∗r (η) = τ

∗r +w

∗dr

(3.12)

with the subscript r as mentionned above denotes the reduced model. When writing ex-

plicitly the above expression, we get:

[M∗z 0

0 M∗ϑ

][z

ϑ

]+

[D∗z 0

0 D∗ϑ

][z

ϑ

]+

[−(W−B)

WzGcos(ϕ)sin(ϑ)

]=

[τ∗z +w

∗dz

τ∗ϑ+w∗dϑ

]

(3.13)

M∗z and M∗

ϑ are the mass parameters in the earth-fixed frame pertaining to z and ϑ

respectively. D∗z andD∗

ϑ are the damping parameters in the earth-fixed frame for these two

degrees of freedom as well. The remaining terms (W, B and zG) are the same as the ones

of equation (3.11) and similarly on the right hand side of the equation, we have the control

inputs summed to the disturbances expressed in the earth-fixed frame for the depth and

the pitch.

3.4 Thruster dynamic modeling

In the dynamic model expressed in (3.6), we find the vector of control input τ ∈ R6×1.

This latter is expressed as a force inNewton for the translational coordinates and a torque

in Newton.meter for the rotational coordinates. When the underwater vehicle is con-

trolled to follow a desired trajectory, the control law calculates the necessary forces and

torques in Newton and Newton.meter to achieve this tracking. Nevertheless, this vec-

tor should be sent to the thrusters. Thrusters for underwater vehicles are usually propellers

driven by electric motors. To achieve a 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 [Blanke et al., 2000]. Most control schemes designed for underwater vehi-

cles tend to ignore the thrusters’ dynamics considering it negligible or classifying it among

the unknown disturbances. During this thesis, a study has been undertaken in Appendix

3.4. THRUSTER DYNAMIC MODELING 41

A with the aim of incorporating this dynamics to stabilize an unactuated. However, it is

important to mention that one of the main obstacles behind a precise modeling of the

thrusters lies in the difficulty to instrument the velocity of the flow. We can therefore de-

duce that the thrust is affected by the motor model and the propeller map [Kim et Chung,

2006]. These two topics are to be discussed here below. For the sake of clarity, Table 3.2

shows the notations used in this section.

Table 3.2: Nomenclature of the notations used in this section

Parameter Description

Ap Propeller disc area

df Quadratic damping coefficient

df0 Linear damping coefficient

Jm Motor/propeller combined moment of inertia

Kn Linear motor damping coefficient

Kn|n| Nonlinear motor damping coefficient

kv1 Back EMF and viscous damping coefficientm Mass of the underwater vehiclemf Mass of the water in the propeller control volumen Propeller shaft speed

Q Propeller torqueτ DC-motor control input

t Thrust deduction number

T Propeller thrustu Surge speed of the vehicleup Axial flow velocity in the propeller discua Ambient water velocity

Xu Linear damping coefficient in surge

Xu|u| Quadratic damping coefficient in surge

Xu Added mass in surge

3.4.1 Propeller shaft speed models

Propeller shaft speed models can be expressed using a one-state model [Yoerger et al.,

1990], a two-state model [Healey et al., 1995] or a three-state model [Blanke et al., 2000]. For


T

Figure 3.2: Schematic view of a propeller with the representation of the vehicle speed u,the axial flow velocity ua, the propeller disk areaAP and the generated thrust T .

a better understanding of these models, the reader can refer to Figure 3.2 for a representa-

tion of a propeller. u is the vehicle speed caused by the thrust T generated by the propeller.

up is the axial flow velocity that usually differs from the vehicle speed and can influence

the thrust at high speeds. The propeller is therefore an actuator disk and the areaAp is the

propeller disk area. In the following, the three types of models will be introduced.

One-state model [Yoerger et al., 1990]

The one-state model is expressed according to the following two equations:

Jmn+Kn|n|n|n|= τ (3.14)

T = f(n,up) (3.15)

Jm is the combined propeller and rotor moment of inertia, n is the shaft rotational speed

in rad.s−1 and n its first derivative,Kn|n| is the nonlinear motor damping coefficient and τ

is the control input (shaft torque) inNewton.meter, (assuming up = 0 (axial flow) when

computing T ). T is the propeller thrust computed through a function f depending on n

and up.

Another one-state model is that proposed in [Bessa et al., 2006] to incorporate some

actuator limitations:

Jmn+kv1n+Kn|nn|n|= τ (3.16)

By incorporating kv1n, the model takes into account the back EMF (Electro Motive Force)

torque and the viscous damping due to mechanical sealing.

3.4. THRUSTER DYNAMIC MODELING 43

Two-state model [Healey et al., 1995]

Similar to the one-state model, the two-state one is also expressed using two differential

equations according to the following:

Jmn+Kn= τ−Q (3.17)

mfup+df(up−u)|up−u|= T (3.18)

In this model, the volume of the water around the propeller was modeled as a mass-

damper system with Kn the linear motor damping coefficient,mf the mass of this control

volume and df the damping coefficient. u is the vehicle speed andQ the propeller torque.

The state variables are the axial fluid velocity up and the propeller rotational velocityn.

Three-state model [Blanke et al., 2000]

The three-state model is expressed by the following three differential equations:

Jmn+Knn= τ−Q (3.19)

mfup+df0up+df|up|(up−ua) = T (3.20)

(m−Xu)u−Xuu−Xu|u|u|u|= (1−t)T (3.21)

This general model includes in comparison to the previous one, the forward dynam-

ics of the vehicle, described by equation (3.17), with m the mass of the vehicle, Xu the

added mass, Xuu the linear laminar skin friction and Xu|u|u|u| the nonlinear quadratic

drag. In (3.16), we notice the use of two coefficients for the damping,namely df0 for the

linear damping and df for the quadratic one.t refers to the thrust deduction number. The

other terms have been defined previously.

3.4.2 Thrust modeling

Modeling the thruster is needed in order to estimate the required voltage that should be

applied to the motor in order to produce the desired force calculated by a control scheme.

For this reason, conventionally, a direct map between the voltage and the thrust force gen-

erated by the controller. If a precise value for the thrust is required, then it is necessary to

have the information concerning the rotational velocity of the propeller. In fact, the pro-

peller thrust is known to depend on the square of its rotational velocity. However, there are

different ways to calculate it. Some of them are listed below:

1. The thrust T produced by marine thrusters which are usually propeller driven can be

calculated as being proportional to the square of the propeller’s rotational velocity


Figure 3.3: Schematic view of the flow model proposed in [Kim et Chung, 2006]

Ωp [Bessa et al., 2006]:

T =CTΩp|Ωp| (3.22)

CT is a coefficient that can be identified experimentally.

In [Whitcomb et Yoerger, 1995], a precise expression of this coefficient is proposed

leading to the following:

T = ρAR2η2 tan(p)2Ωp|Ωp| (3.23)

where ρ is the ambient fluid density,A is the propeller area,R the winding resistance,

η the propeller efficiency and p the fluid pressure. This model was used for compar-

ison with a simpler model the authors proposed where rotational propeller data and

motor dynamics where omitted.

2. In [Bessa et al., 2006] actuator limitations were added to the previous model:

T =D(Ωp|Ωp|) (3.24)

D(Ωp|Ωp|) represents the dead zone nonlinearity with the quadratic inputΩp|Ωp|

and output T .

3. A complete different concept for calculating the thrust of the propeller is the one

based on lift and drag curves using thin airfoil theory [Fossen et Blanke, 2000]. The

lift and drag are usually represented as nondimensional thrust and torque coeffi-

cients computed from self propulsion tests:

T = ρD4KT (J0)Ωp|Ωp| (3.25)

KT is the thrust coefficients that is a linear combination of the advance ratio J0 being

the ratio between the axial flow velocity and the product of the propeller shaft with

the propeller diameter. D is the diameter of the propeller and ρ the ambient fluid

density.

3.5. CONCLUSION 45

4. Based on Bernouill’s equation, in [Kim et Chung, 2006] the thrust coefficient is also

computed using hydrodynamic relations:

T = 2ρAp(k1u+k2DΩp)(k1u+k2DΩp−u) (3.26)

up= k1u+k2DΩp (3.27)

k1 and k2 being are constants used for the calculation of the axial flow as shown in

(3.23) and Figure 3.3. The other variables have been already described above.

5. Bachmayer in [Bachmayer et al., 2000] presented a thrust model incorporating the

effects of the rotational fluid velocity and the inertia on the thruster’s response and

determined the nonsinusoidal lift/drag forces through hybrid simulations.

In this thesis, the model as defined in [Bessa et al., 2006] and expressed in (3.22) will

be used. In fact, it was implemented in the study of the roll stabilization presented in

appendix A. This model has been chosen for the simplicity of its implementation. In fact,

most of the other methods require coefficients to be computed from self propulsion tests

as well as additional knowledge concerning other variables and parameters (AP, η, ect.).

3.5 Conclusion

In this chapter the modeling of an underwater vehicle was presented. A model is impor-

tant for simulation purposes in order to get an idea of the behavior of the system in closed-

loop. For this reason, the motion variables have been defined in the earth-fixed frame and

the body-fixed frame in order to establish the dynamic model that was presented using

the formulation described in [Fossen, 2002]. In order to ensure the mapping between the

thrust force and the required input voltage, thrust modeling is required. Different models

exist for this mapping and the one presented in [Bessa et al., 2006] was chosen for the study

of the roll stabilization shown in Appendix A.

Part II

Proposed Solutions

47

CHAPTER

4Solution 1: Conventional controllers

The conventional view serves to

protect us from the painful job of

thinking

JOHN KENNETH GALBRAITH

Contents

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 PID control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 Nonlinear adaptive state feedback control . . . . . . . . . . . . . . . . . . . 53

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1 Introduction

In this chapter two controllers already used in underwater robotics are proposed for

depth and pitch control. We qualified them as conventional given their popularity and

common use in the field of underwater robotics. The two controllers to be described

hereby are the PID controller and the nonlinear adaptive state feedback one. The former

controller is the classical one explained in any basic control theory textbook but whose pa-

rameters (feedback gains) have been computed using a specific tuning method. The latter

controller is the one described in [Fossen, 2002] to be applied on an underwater vehicle.

A theoretical background on each of these two control schemes will be provided before

addressing their design methods for the depth and pitch control of an underwater vehicle.

49

50 CHAPTER 4. SOLUTION 1: CONVENTIONAL CONTROLLERS

+− +++

P

I

D

∫t

0I dt)t(eK

dt

)t(deKD

)t(eKP

)t(u)t(e )t(y)t(r

Figure 4.1: Block diagram of the PID based control loop.

4.2 PID control

4.2.1 Background

The Proportional Integral Derivative (PID) control (cf. Figure 4.1) can be described by

the following equation:

u(t)=Kpe(t)+Ki

∫ t

0

e(t)dt+Kdde(t)

dt(4.1)

with e(t) being the error signal given by e(t) = r(t)−y(t), with r(t) the reference trajec-

tory and y(t) the output to be controlled. In our case, the output the depth and pitch.

Kp is the proportional gain, Ki the integral gain and Kd the derivative one. Increasing the

proportional gain would be to compromise the transient behavior, given the emerging os-

cillations and overshoots, at the expense of a faster response with a reduced steady state

error. The integral action aims at eliminating the steady state error but might induce in-

stability when it is increased. Finally the derivative gain increases the system damping to

reduce the oscillations and overshoots and hence to improve the stability of the system.

Equation (4.1) can be rewritten as:

u(t)=Kp

(e(t)+

1

Ti

∫ t

0e(t)dt+Td

de(t)

dt

)(4.2)

4.2. PID CONTROL 51

y(t)

t

a

L

Figure 4.2: Graphical parameter estimation of an integrator model

In this case Ti is called the integral time and Td the derivative time. This description of

the controller will be useful for determining the gains to adequately tune the PID controller.

4.2.2 PID Controller Design

To tune the parameters of a PID controller, several methods exist in the literature such

as Ziegler-Nichols tuning method, Chien-Hrones-Reswick formula, etc . These methods

can be found in any control textbook such as [Visioli, 2001]. Two tuning methods will be

presented and tested. Both use the information extracted from the open-loop step re-

sponse. The first one calculates the tuning parameters after approximating the vehicle’s

model by a known available one. And the second one (Ziegler Nichols) directly calculates

the parameters from some variables graphically extracted.

PID parameters for an integrator plus dead time model

With an a priori knowledge of the dynamics and the output response of our plant, we

can approximate the depth behavior of our underwater vehicle by an Integrator Plus Dead

Time (IPDT) model given by:

G(s) =a

sLe−sL (4.3)

where the parameters L and a can be defined to be the intersections of the tangent to the

system step response with the x and y axes respectively as illustrated in the Figure 4.2, and

s is the Laplace variable.

With these two parameters in hand, various controllers can be designed according to


Table 4.1: Coefficients of the PID controller for the integral plus dead time plants

criterion a1 a2 a3 a4 a5

ISE 1.03 0.49 1.37 1.49 0.59

ITSE 0.96 0.45 1.36 1.66 0.53

ISTSE 0.9 0.45 1.34 1.83 0.49

some specific selected criterion. In our case, we chose to design our controller in order

to minimize the ISTSE (Integral of Squared Time multiplied by Squared Error) based on

the algorithm described in [Visioli, 2001]. The coefficients pertaining to this selected al-

gorithm are extracted from Table 4.1. This table holds all the coefficients for the design of

either a PD or a PID according to different criteria (Integral of Squared Error (ISE), Integral

of Time Squared Error (ITSE) and Integral of Squared Time multiplied by Squared Error

(ISTSE)). The parameters a1 and a2 are used for the design of a PD whereas a3, a4 and a5are used for a PID. The identification of the PID feedback gains becomes straightforward.

With a3= 1.34 , a4= 1.83, and a5= 0.49 and by lettingK= aL , the gains are found to be:

Kp=a3KL Ki=

KpTi

Kd=KpTd

with Ti=a4L and Td=a5L.

Ziegler Nichols method in open-loop

y(t)

t

Figure 4.3: Graphical illustration of the step response

The parameter setting used for the IPDT model presented above works only for the

depth since the pitch of the underwater vehicle does not follow the same dynamics. For

this reason, a different tuning method should be used. For instance, the Ziegler Nichols

4.3. NONLINEAR ADAPTIVE STATE FEEDBACK CONTROL 53

method for tuning in open-loop can be used. A step response is firstly performed on the

pitch angle and the parameters Ta and Tu are extracted graphically as illustrated in Figure

4.3. The parameters of the PID are therefore calculated according to the following:

Kp= 1.2TaTu, Ti= 2Tu, Td= 0.5Tu

The Ki and Kd gains can be deduced as stated before as: Kd=KpTi, Kd=KpTd

4.2.3 Application for depth and pitch control

A classical PID controller as described above has been used for the tracking control of

the depth and pitch. To control these two degrees of freedom, our control input is therefore

of dimension 2 according to the following:

τ=

[τzdesτϑdes

](4.4)

withτz=Kpz(zdes−z)+Kiz

∫t0(zdes−z)dt−Kdz(zdes− z)

τϑ=Kpϑ(ϑdes−ϑ)+Kiϑ∫t0(ϑdes−ϑ)dt+Kdϑ(ϑdes− ϑ)

(4.5)

where τz is the input in Newton to be applied along the z axis, Kpz ,Kiz and Kdz are

positive constants gains representing respectively the proportional, the integral and the

derivative gains of the first PID controller. z is the measured depth, zdes the desired one

and z and zdes are their respective time derivatives. τϑ is the input in Newton.meter to

be applied for pitch control. Kpϑ,Kiϑ and Kdϑ are the feedback gains of the second PID

controller. ϑ and ϑdes are the measured pitch and the desired one and ϑ and ϑdes their

respective derivatives.

4.3 Nonlinear adaptive state feedback control

4.3.1 Background

The adaptive state feedback controller is capable of adjusting over time under changes

occurring in the model parameters or the environment. This scheme provides an online

estimation of the unknown model parameters in order to ensure a good trajectory track-

ing for the closed-loop system [Fossen, 2002]. The obtained control law is based on the

dynamics of the robot presented in 3.7 and it is given by:

τ= Mab+ n(ν,η) (4.6)


θηνΦ ˆ),,a( b

∫−−−t

IPDd dtKKK0

~~~ ηηηη &&&

)Ja(J n1 ν&−−

yJ),,a( 1bT −− ηνΓΦ ∫

parameter update control law

commanded accelera!on in the earth frame

commanded

accelera!on in

the body frame

na

ba

θ& θ τ

d

d

η

η

&

ηη&

)t(),t( νη

ROV

+−

η

η&~

~

Figure 4.4: Block diagram of the nonlinear adaptive state feedback controller

where the M denotes the mass estimate, ab the commanded acceleration described

in the body-fixed frame, and n(ν,η) the estimate of the nonlinear part n(ν,η) where

n(ν,η) = D(ν)ν+g(η).

Given that the dynamic model is linear in its parameters, the adaptive control law (4.6)

can then be rewritten as:

τ=Φ(ab,ν,η) θ (4.7)

where Φ is the regressor and θ is the vector of the estimated parameters. The computed

input is calculated in the body-fixed frame while the trajectory following is performed in

the earth-fixed frame. Therefore, ab is calculated from a transformation between the body

and the earth fixed frames and is given by:

ab= J−1(an− Jν) (4.8)

where an is the commanded acceleration in the earth-fixed frame and J is the transforma-

tion matrix from the body-fixed frame to the earth-fixed frame with J being its first deriva-

tive. To ensure that the tracking error converges to zero, an is chosen as follows:

an= ηdes−Kpη−Ki

∫ t

0

ηdt−Kd ˙η (4.9)

4.3. NONLINEAR ADAPTIVE STATE FEEDBACK CONTROL 55

with η= η−ηdes is the tracking error and ˙η is its first time derivative, ηdes is the desired

trajectory and ηdes is its corresponding acceleration. Kp, Ki and Kd are the proportional,

integral and derivative gains respectively.

The vector of the estimated parameters is adapted according to the following update law:

˙θ=−ΓΦT (ab,ν,η)J−1yA (4.10)

where Γ is a diagonal positive definite matrix representing the adaptation gain, and yA is

the combined error defined as follows:

yA= c0η+c1 ˙η (4.11)

c0 and c1 are constant positive gains. The choice of their values is governed by the algo-

rithm presented in [Fossen, 2002] guaranteeing the convergence of the error to zero. The

proof of the closed-loop stability is made by applying Barbalat’s lemma and is explained in

appendix B.

4.3.2 Application for depth and pitch control

The vector of parameters to be estimated includes elements of the matricesM andD

and g described in section 3.3. In the following equations ranging from (4.12) to (4.18), we

explicitly formulate our controller to apply it on the reduced model related to the body-

fixed frame seen in section 3.3 in equation (3.11). We therefore get:

τr=Φrθr (4.12)

with the vector of the estimated parameters being:

θr=[Mz Dz W−B Mϑ Dϑ zGW

]T(4.13)

Mz and Dz are the estimates of the mass and damping parameters along the z axis. W−B

is the estimate of the parameter representing the difference between the weight and the

buoyancy. Mϑ and Dϑ are the estimates of the rotational mass and damping parameters

around the y axis, zG is the z coordinate of the center of gravity and zGW is the estimate of

the gravitational parameter pertaining to the pitch angle.

Having defined the parameter vector, the regressor takes the form:

Φr=

[abz w −cos(ϑ)cos(ϕ) 0 0 0

0 0 0 abϑ q sin(ϑ)

](4.14)


the commanded acceleration in the earth-fixed frame is given by:

anr =

[zdes

ϑdes

]−Kp

[z−zdes

ϑ−ϑdes

]−Ki

∫ t

0

[z−zdes

ϑ−ϑdes

]dt−Kd

[z− zdes

ϑ− ϑdes

](4.15)

Kp,Ki and Kd ∈ R2×2 are diagonal positive definite matrices representing respectively the

proportional, the integral and the derivative gains. According to (4.8), the commanded

acceleration in the body-fixed frame is given by:

abr =

[cos(ϕ)cos(ϑ) 0

0 cos(ϕ)

].

(anr−

[−qsin(ϑ)−pcos(ϑ)sin(ϕ) 0

0 −ϕsin(ϕ)

][w

q

])

(4.16)

the adaptation law (4.10) is written as:

˙θr=−ΓrΦTr

[cos(ϕ)cos(ϑ) 0

0 cos(ϕ)

]yAr, (4.17)

and the combined error (4.11) is expressed by:

yAr = c0

[z−zdes

ϑ−ϑdes

]+c1

[z− zdes

ϑ− ϑdes

]. (4.18)

It is important to emphasize that a the convergence to the estimated parameters is

more guaranteed when the reference trajectory is rich enough to excite the parameters

to be estimated [Slotine et Weiping, 1991]. These parameters will converge to a set of val-

ues that allows trajectory following. However, it is worth to notice that this set of values

does not necessarily need to fit with the real values of the parameters but with a set of val-

ues allowing the convergence. Moreover, the vector of parameters is seen to be bounded

according to Barbalat’s lemma as shown in the proof of stability given by [Fossen, 2002].

4.4 Conclusion

In this chapter two conventional controllers (PID and nonlinear adaptive state feed-

back) have been presented and designed for depth and pitch control of an underwater

vehicle. The PID controller can be useful for systems where the plant model is unknown

since it is non model based but, as we will see later, tuning its gains remains a challenge.

The adaptive controller can estimate the unknown parameters of the plant in order to con-

verge to the desired trajectory. In fact, unlike the PID, it is capable of compensating for

variations and uncertainties in the model thanks to this auto tuning ability. However, the

parameter vector needs to be initialized with some values close to the real ones in order to

ensure a good transient behavior and a fast adaptation time.

CHAPTER

5Solution 2: Nonlinear L1 adaptive

controller

Adapt or perish, now as ever, is

nature’s inexorable imperative.

H. G. WELLS

Contents

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 From MRAC to L1 adaptive control . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Background on L1 adaptive control . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 State feedback L1 controller from nonlinear multi-input systems with

uncertain input gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.5 Design of a multi-variable controller for depth and pitch control in un-

derwater robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1 Introduction

Adaptive control schemes are seen to be very common and useful in underwater

robotics [Marzbanrad et al., 2011][Bessa et al., 2008][Antonelli et al., 2001]. The use of an

adaptive controller, as seen in the previous chapter, is motivated in particular by the pres-

ence of uncertainties in the model parameters and their likelihood to change. However,

despite the success of such techniques in many applications, they hold some drawbacks.

57

58 CHAPTER 5. SOLUTION 2: NONLINEAR L1 ADAPTIVE CONTROLLER

For instance, adaptive controllers rely on the need of a persistency in parameter excitation

before convergence which may lead to a bad transient behavior [Zang et Bitmead, 1990].

A regressor is often required, involving with it a large parameter vector to be estimated.

Moreover, a large adaptation gain may have undesirable effects, with the risk of parame-

ter divergence. All of the arguments brought against adaptive schemes reveal that, despite

their numerous advantages, these controllers hold some drawbacks. For the sake of clarity,

we can summarize some of them here:

i) A wide range of such controllers exhibit undesirable frequency characteristics and are

often used with restrictive assumptions. In [Rohrs et al., 1982], the authors showed

that sinusoidal reference inputs at certain frequencies and/or sinusoidal output dis-

turbances at any frequency will cause the adaptation gain to significantly increase,

which will destabilize the control system.

ii) The need for the persistency in excitation can lead to a bad transient behavior [Zang

et Bitmead, 1990].

iii) An increase in the adaptation gain drives the closed-loop system closer to instability,

while a small gain would slow down the convergence [Narendra et Annaswamy, 1987].

iv) Any parameter vector to be adapted must be adequately initialized and this choice

would depend on the specific configuration of the system. This would be even more

critical for the non-model-based controllers [Antonelli, 2007].

That is why a control approach that could ensure a robustness decoupled from adap-

tation would be highly desirable. Such a scheme would drive the closed-loop system to

its desired trajectory, while overcoming the drawbacks listed above. This decoupling has

recently been proven to be achievable by the L1 adaptive control scheme presented in

[Hovakimyan et Cao, 2010]. This method can be compared to a Model Reference Adaptive

Controller (MRAC) modified in two ways: a state predictor is used instead of a reference

model, and a low-pass filter is inserted in the feedback loop to cancel out undesirable high

frequencies in the control input. This controller can ensure a good performance with zero

parameter initialization and without any necessity for a specific excitation. The proof of

asymptotic stability of this controller is performed through the small gain theorem [Ho-

vakimyan et Cao, 2010]. In summary, this control scheme is able to revisit the failures of

other adaptive controllers by maintaining its performance and robustness in situations

where the other controllers cannot [Kharisov et Hovakimyan, 2010][Xargay et al., 2009].

The L1 adaptive control has been validated through numerical simulations and real-time

experiments mainly on aerial vehicles [Dobrokhodov et al., 2010][Kaminer et al., 2010], but

it was also seen in other applications such as the control of the acrobot [Techy et al., 2007],

the hysteresis in smart materials [Fan et Smith, 2008], and the regulation of arterial CO2

5.2. FROM MRAC TO L1 ADAPTIVE CONTROL 59

tension in blood [Promprara et al., 2013]. In this chapter, a quick overview on MRAC con-

trol will be given to introduce with it the concept and motivation behind the establishment

of the L1 adaptive controller. A linear time invariant system in presence of uncertain con-

stant parameters is later on considered in order to explain how this particular architecture

is constructed. Finally, a nonlinear state feedback L1 adaptive controller for multi-input,

multi-output nonlinear systems will be presented. It will be used for the design of a depth

and pitch controller for an underwater vehicle. This leads us to the main contribution of

this chapter: the formulation of a novel scheme to suit a new field of application, which is

underwater robotics.

5.2 From MRAC to L1 adaptive control

In this section two different versions of MRAC will be presented. Despite the presence

of a variant in their architectures, the same error dynamics is reached. However, this vari-

ant is considered to be one of the main elements that lead to the establishment of L1 adap-

tive control.

5.2.1 From direct MRAC to direct MRAC with state predictor

Direct MRAC

Let us consider the following linear time invariant system given by:

x(t)=Amx(t)+b(u(t)+kTxx(t)), x(0)= x0

y(t) = cx(t)(5.1)

x(t) ∈ Rn is the measured state of the system and Am ∈ Rn×n is a Hurwitz matrix defining

its closed-loop dynamics. b ∈ Rn is a known constant, u(t) ∈ R is the control input and

kTx ∈Rn is the vector of unknown constant parameters. The output y ∈ R is defined thanks

to the constant vector c ∈Rn being multiplied by the system’s state vector x(t).

Having defined a desired reference trajectory r(t), the control objective resides in the de-

sign of an adaptive state feedback input u(t), so that the output y(t) tracks r(t) while

preserving the boundedness of the states and parameters of the system.

As specified in its nomenclature, a scheme based on MRAC uses a reference model. To

define this latter, we start by setting the constant kg =−1

cA−1m b

. The choice of this constant

ensures that the reference input is being tracked with zero steady state error. The nominal

controller can be therefore defined according the following:

unom(t)=−kTxx(t)+kgr(t) (5.2)


unom has been defined as such in order to cancel the uncertainties present in the system

(5.1) (being the unknown vector kx).

With the above definition of the input used in our system (5.1), we construct the refer-

ence model and we express it according to the following:

xm(t)=Amxm(t)+bkgr(t), x(0) = x0

ym(t)= cxm(t)(5.3)

xm ∈Rn and ym ∈R are respectively the states of the reference model and its output.

Having defined the reference model (5.3), the control input u(t) to be applied on the

system such that its response matches the one of the reference model is:

u(t)=−kTx(t)x(t)+kgr(t) (5.4)

with kTx(t)∈Rn being the estimate of kx at each instant t.

The tracking error e(t) = xm(t)−x(t) is the difference between the state of the refer-

ence model and the measured one. In order for this error to converge to zero, the following

adaptation law to estimate kx is defined:

˙kx(t)=−Γx(t)eT (t)Pb (5.5)

Where ˙kx is the time derivative of kx, the estimate ofkx. kx is a priori initialized with a value

chosen according to some preliminary knowledge of the plant. Γ ∈ R+ is the adaptation

gain, and P=PT > 0 is the solution for the algebraic Lyapunov equation:

ATmP+PAm=−Q

Q being an arbitrary matrix such that Q=QT > 0. With a suitable choice of a Lyapunov

function, and by applying Barbalat’s lemma, the tracking error is proven to asymptotically

converge to zero. However, the error of the parameter estimation (kx(t) = kx(t)−kx) is

only guaranteed to stay bounded. The block diagram of the closed loop system is displayed

in Figure 5.1. This figure summarizes the architecture of the MRAC controller defined by

four blocks: the reference model to be matched, the controlled system, the adaptation

law estimating the uncertainties, and the control law. From this scheme, we introduce the

direct MRAC with state predictor explained in the next section. The reference model to be

matched will be replaced with a state predictor.

5.2. FROM MRAC TO L1 ADAPTIVE CONTROL 61

5.2.2 From direct MRAC with state predictor to L1 adaptive control

We consider the same system defined in (5.1) and we formulate its state predictor ac-

cording to the following:

˙x(t)=Amx(t)+b(u(t)+ kTx(t)x(t)), x(0)= x0

y(t)= cx(t)(5.6)

x(t)∈Rn is the state of the predictor of the system and kTx(t)∈Rn is the estimate of the

vector of unknown constant parameters. Finally y ∈R is the estimate of the output.

In the previous section for the direct MRAC, the tracking error e(t) was defined. In this

case, it is replaced with the prediction error ˜x(t) = x(t)−x(t). However, the same control

input defined in (5.4) is used but the adaptation law for the estimation of the uncertain

parameter kx is modified and it becomes:

˙kx(t)=−Γx(t)xT (t)Pb (5.7)

Γ ∈R+ is the adaptation gain, and P= PT > 0 is the solution for the algebraic Lyapunov

equation:

ATmP+PAm=−Q (5.8)

Q being an arbitrary matrix such that Q=QT > 0. When comparing this adaptation

law with the one expressed in (5.5), we notice that the tracking error e(t) was replaced

by the prediction error x(t). The error dynamics of these two MRAC methods are in fact

identical, and the proof of stability for the latter case is formulated the same way as for the

former one. The same Lyapunov function is used by replacing e(t) by x(t). For a complete

derivation of the proof of stability, the reader can refer to [Hovakimyan et Cao, 2010]. In this

reference, the authors prove that the tracking (or prediction) error at any time t is upper

bounded by:

||e(t)||or ||x(t)||≤||kx(0)||√λmin(P)Γ

(5.9)

kx(0) = kx(0)−kx(0) is the initial estimation error on the parameter kx, λmin(P) is

the minimum eigen value of the matrix P defined above as the solution to the Lyapunov

equation (5.8).

The above equation shows that the tracking or prediction error can go to zero faster

when large adaptation gains are used. However, as seen in (5.5) and in (5.2), this adapta-

tion gain enters in the update law. Therefore, setting it to large values induces high gain

feedbacks leading to the instability of the system.


9

-

+

Reference Model

System

Adaptation Law

Control Law

Figure 5.1: Block diagram of the control loop based on the direct MRAC architecture

9

-

+

State Predictor

System

Adaptation Law

Control Law

Figure 5.2: Block diagram of the control loop based on the direct MRAC architecture withstate predictor

9

-

+

State Predictor

System

Adaptation Law

Control Law

Figure 5.3: Block diagram of the control loop based on the L1 adaptive controller

In conclusion, increasing the adaptation gain would improve the tracking but in the

same time would hurt the robustness of the closed-loop system. It is a trade off between

performance and robustness. L1 adaptive control was born from this premise. It is based

5.3. BACKGROUND ON L1 ADAPTIVE CONTROL 63

on an architecture where robustness and adaptation are decoupled. The four blocks used

for the direct MRAC with state predictor and displayed in Figure 5.2 were also used for

the establishment of theL1 controller. The added difference is that the control input is low

pass filtered byC(s) as seen in Figure 5.3. This novel scheme is detailed in the next section.

5.3 Background on L1 adaptive control

As mentionned previously, the choice of this controller is motivated by its specific

architecture characterized by the decoupling between adaptation and robustness. High

adaptation gains can then be chosen ensuring a fast convergence with a smooth transient

response. Its basic architecture, described in [Hovakimyan et Cao, 2010], is illustrated in

the block diagram of Figure 5.4. The closed-loop system holds a prediction phase and an

adaptation phase. In the feebdack, a low pass filter is added to cancel out the high fre-

quencies that might occur in the control input. This last component can ensure a fast

adaptation without altering the robustness. In the following, a state feedback L1 controller

will be presented using a system with constant uncertain parameters as seen in section

1.2. However, the control problem will be presented according to the formalism of the L1

controller where minor changes in the formulation of the system and control are brought,

and compared to the direct MRAC with state predictor.

Figure 5.4: Block diagram of the closed-loop L1 adaptive controller

5.3.1 State feedback L1 controller for linear time invariant systems

The details of the different blocks of Figure 5.4 are presented in the following:

– Controlled System: We will start by considering the following class of systems:

x(t)=Ax(t)+b(u(t)+θTx(t)), x(0)= x0

y(t)= cx(t)(5.10)


x(t)∈Rn is the measured state of the system andA∈Rn×n is a known matrix not nec-

essarily Hurwirtz but conforms with the condition (A,b) being controllable. b ∈ Rn

is a known constant,u(t)∈R is the control input and θ ∈Rn is the vector of unknown

constant parameters with a known bound. The output y ∈R is defined thanks to the

constant vector c ∈R1×n being multiplied by the system’s state vector x(t).

Having defined a desired reference trajectory r(t), the control objective resides in

the design of an adaptive state feedback input u(t) so that the output y(t) tracks

r(t) while preserving the boundedness of the states and parameters of the system.

This control input is composed of two parts defined according to the following:

u(t)=um(t)+ua(t), with um(t)=−kmx(t) (5.11)

um is the component rendering the matrix A Hurwitz thanks to km ∈ R1×n being

the static feedback gain that transforms A into Am = A−bkm. The matrix Am is

therefore the one delimiting the closed-loop dynamics of the system. This leaves us

with ua being the adaptive control input to be designed.

By combining (5.10) and (5.11), we get:

x(t)=Amx(t)+b(ua(t)+θT (t)x(t)), x(0) = x0

y(t)= cx(t)(5.12)

The matrix A is replaced by the Hurwitz one Am thanks to the incorporation of

um(t). The control input is therefore reduced to the adaptive component ua(t).

– State predictor: The states of the system are computed at every iteration based on

the estimated parameters obtained from the adaptation phase along with the control

input. The predictor is therefore constructed according to the following:

˙x(t)=Amx(t)+b(ua(t)+ θT (t)x(t)), x(0) = x0

y(t)= cx(t)(5.13)

x(t) ∈ Rn is the state of the predictor of the system and θT (t) ∈ Rn is the estimate

of the vector of unknown constant parameters. Finally y ∈ R is the estimate of the

output.

– Adaptation: This phase uses the error between the measured and the estimated

states to adapt the parameters together with a projection method in order to ensure

their boundedness. Indeed, a projection operator avoids the parameter drift using

the gradient of a convex function and a maximal bound on the parameters to be es-

timated. The adaptation law for each estimated parameter is then given by:

˙θ(t)=ΓProj(θ(t),−xT(t)Pbx(t)) (5.14)

5.3. BACKGROUND ON L1 ADAPTIVE CONTROL 65

9

-

+

Adaptive Controller

-

+

Figure 5.5: Block diagram of the control loop based on the L1 adaptive controller

The parameterP is the solution of the algebraic Lyapunov equation:ATmP+PATm = −Q

for any arbitrary symmetric matrixQ=QT > 0. Γ is the adaptation gain and x(t) the

error between the predicted and the measured states.

– Control law with low pass filter: The last stage pertains to the adaptive component of

the control input characterized by the consideration of a low pass filter. It is written

in Laplace form as:

ua(s) =−C(s)(ηl(s)−kgr(s)) (5.15)

where C(s) is a bounded input, bounded output stable and strictly proper transfer

function, ηl(t) = θT (t)x(t), kg =− 1

cA−1m b

and r(s) the Laplace transform of the ref-

erence trajectory r(t).

The system (5.12) under the L1 adaptive controller proposed by (5.15) is guaranteed to be

bounded input bounded state stable with respect to reference trajectory and initial condi-

tions if km andC(s) verify with following L1 norm condition:

||G(s)||L1L< 1 (5.16)

where

G(s) =H(s)(1−C(s)), H(s) = (sI−Am)−1b, L=max||θ|| (5.17)

G(s) and H(s) and C(s) being bounded-input bounded-output stable transfer func-

tions. L is the maximal bound set on the parameter θ.

Figure 5.5 shows the block diagram of the control loop based on theL1 controller when

applied on a linear system with a constant unknown parameter vector.

Remark 1: This remark concerns the role of the input component um.


From the basic example given above, it is possible to extend this controller to different

classes of systems. We briefly mention here below a system with an uncertain input gain

and time-varying parameters.

x(t)=Ax(t)+b(ωu(t)+θT(t)x(t)+σ(t)), x(0)= x0

y(t) = cx(t)(5.18)

Similarly to the model (5.10),x(t) ∈Rn is the measured state of the system andA∈Rn×n

is a known matrix not necessarily Hurwirtz but conforms with the condition (A,b) being

controllable. b ∈ Rn is a known constant, u(t) ∈ R is the control input and θ ∈ Rn is the

vector of unknown time-varying parameters. The output y ∈ R is defined thanks to the

constant vector c ∈ R1×n being multiplied by the system’s state vector x(t). Two new ele-

ments are added: ω ∈R being the uncertainty on the input gain and σ(t) ∈R representing

the input disturbances.

u(t) takes the form :

u(t)=um(t)+ua(t) (5.19)

with um =−kmx(t). um as seen previously is used to transform A into a Hurwitz matrix

Am.

We set g0(t,x(t))= θT (t)x(t)+σ(t). Here below will be detailed the substitution of the

control law (5.19) into (5.18) :

x(t)=Ax(t)+b(ω(ua(t)+um)+g0(t,x(t))

x(t)=Ax(t)+b(ωua(t)−ωkmx(t)+g0(t,x(t))

x(t)=Ax(t)+b(ωua(t)−ωkmx(t)+kmx(t)−kmx(t)+g0(t,x(t))

x(t)= (A−bkm)x(t)+b(ωua(t)−ωkmx(t)+kmx(t)+g0(t,x(t))

x(t)=Amx(t)+b(ωua(t)+kmx(t)(1−ω)+g0(t,x(t))

x(t)=Amx(t)+b(ωua(t)+g1(t,x(t))

with g1(t,x(t))=g0(t,x(t))+kmx(t)(1−ω).

From the derivations, we notice thatum was not the one to renderAHurwitz but it was

essential in transformingg0 into a new modifiedg1. When an uncertainty on the input gain

is present, care should be taken when designingAm. Given thatω is unknown, it cannot

enter in the formulation of the matrix Am. For this reason, ω should be normalized in

order to ensure a nominal case aroundω= 1. Failing to do so might increase the value of

g1(t,x(t)) making the condition of stability hard to satisfy.

It is important to mention that the component um is not compulsory. It is possible

to directly set a state matrix Am Hurwitz by specifying certain poles to satisfy a desired

5.4. STATE FEEDBACK L1 CONTROLLER FROM NONLINEAR MULTI-INPUT SYSTEMSWITH UNCERTAIN INPUT GAINS 67

dynamics. However, going from a matrixA can be advised in certain cases when a certain

knowledge of the plant dynamics is known. This will help in reducing the bounds on the

parameters and terms to be estimated.

5.4 State feedback L1 controller from nonlinear

multi-input systems with uncertain input gains

5.4.1 Problem formulation

– Controlled System: We will start by considering the following general class of nonlin-

ear systems which includes underwater vehicles:

x1(t)= x2(t), x1(0)= x10x2(t)= f2(t,x(t))+ωB2u(t), x2(0)= x20y(t)=Cx(t)

(5.20)

where x1(t) ∈ Rn and x2(t) ∈ Rn are the states of the system constituting the com-

plete state vector: x(t) = [xT1 (t), xT2 (t)]

T . u(t) ∈ Rm is the control input vector

(m ≤ 2n) and ω ∈ Rm×m is the uncertainty on the input gain. B2 ∈ Rn×m is a con-

stant full rank matrix. C ∈ Rm×2n is a known full rank constant matrix, y ∈ Rm is

the measured output and f2(t,x(t)) is an unknown nonlinear function represent-

ing the nonlinear dynamics. The partial derivatives of this function are assumed

to be semiglobally uniformly bounded and f2(t,0) is assumed to be bounded. The

previous system of equations (5.20) can be transformed into a semi-linear one as

described in [Cao et Hovakimyan, 2008]. It is concluded that this function can be

rewritten as f2(t,x(t)) =A2x2+θ(t)||x(t)||L∞+σ(t). A2 ∈ R

n×n is the state matrix

representing the linear part of the dynamics of x2, θ(t) ∈ Rn is the vector of uncer-

tain parameters and σ(t) ∈ Rn is the vector the lumped unknown nonlinear terms.

||x(t)||L∞is the infinity norm of the measured state at each iteration. Therefore the

system can be expressed according to the following:

x(t)=

[0n×n In

0n×n A2

] [x1

x2

]+

[0n×1

θ

]||x||L∞

+

[0n×1

σ

]+

[0n×m

B2

]ωu(t)

y(t)=Cx(t)(5.21)

Let A =

[0n×n In

0n×n A2

]be the state matrix describing the actual open-loop system

dynamics. It should be modified into a Hurwitz matrixAm with the desired closed-

loop dynamics using a static feedback gain km ∈ Rm×2n. We would therefore get


Am = A−Bmkm, with Bm =

[0n×m

B2

]. The final control input to be applied to

the system isu=um+ua withum=−kmx. ua is the control input used for adapta-

tion after the transformation of the matrixA intoAm by um as explained in section

5.3.1. The system can then be finally rewritten in the following compact form as:

x(t)=Amx(t)+Bm

(ωua+θ(t)||x(t)||L∞

+σ(t)), x(0) = x0

y(t)=Cx(t)(5.22)

Given their structure, the vectors θ and σ can be summed to the control input as

shown above. In case the vector B2 is not an identity matrix, these two uncertain

varying parameters would be scaled by the constants contained in B2.

– State predictor: We construct the predictor pertaining to the controlled system (5.22):

˙x(t)=Amx(t)+Bm(ω(t)ua(t)+ θ(t)||x(t)||L∞

+ σ(t))

(5.23)

– Adaptation: As seen in the previous section, this phase uses the error between the

measured states and the estimated ones to adapt the parameters using a projection

method. Three parameters here are to be adapted:

˙θ(t)=ΓProj(θ(t),−(xT(t)PBm)T ||x(t)||L∞

)

˙σ(t)=ΓProj(σ(t),−(xT(t)PBm)T )

˙ω(t)=ΓProj(ω(t),−(xT(t)PBm)TuTa(t))

(5.24)

The parameterP is the solution of the algebraic Lyapunov equation:ATmP+PATm = −Q

for any arbitrary symmetric matrixQ=QT > 0. Γ is the adaptation gain and x(t) the

error between the predicted and the measured states.

– Control law with law pass filter: The last stage pertains to the formulation of the con-

trol input characterized by the addition of a low pass filter. It is written as:

ua(s) =−kD(s)(ηl(s)−kgr) (5.25)

D(s) is an m×m strictly proper transfer matrix leading to the stable closed-loop

filter: C(s) = ωkD(s)Im+ωkD(s)

. k is a positive feedback gain, kg=−(CA−1m Bm)

−1 is a feed-

forward prefilter applied to the reference signal r(t) and ηl= ω(t)ua(t)+θ||x(t)||L∞.

To ensure stability, the feedback gain k and the filter D(s) must be chosen in order

to fulfill the L1 norm condition. This condition and the stability analysis will be de-

tailed in next section. The reader can refer to [Hovakimyan et Cao, 2010] for more

details about the complete proof of stability.

The overall control architecture with the equations included can be summarized in the

block diagram of Figure 5.6.

5.4. STATE FEEDBACK L1 CONTROLLER FROM NONLINEAR MULTI-INPUT SYSTEMSWITH UNCERTAIN INPUT GAINS 69

99 9

-

+

Ad

ap

tive

Co

ntr

olle

r

-+9

Figure 5.6: Block diagram of the control-loop based on the L1 adaptive control scheme


5.4.2 Stability Analysis

To prove the stability of the system (5.20), an inequality explained here below is to be

satisfied under some assumptions.

To formulate these assumptions, we encompass the uncertain and nonlinear terms in

one function g(t,x(t)). Therefore the system (5.20) put under the compact form (5.22) can

be written according to the following:

x(t)=Amx(t)+Bmωu(t)+g(t,x(t)), x(0) = x0

y(t)=Cx(t)(5.26)

where g(t,x(t)) = [gT1 (t,x(t)), gT2 (t,x(t))]T is the nonlinear function with the un-

certain parameter vector θ and the nonlinear terms σ. In our specific case and accord-

ing to (5.20), we have g1(t,x(t)) = 0n, and g2(t,x) = [gT21(t,x), gT22(t,x), ...gT2n(t,x)]

T =

Bm(θ(t)||x(t)||L∞+σ.

The considered assumptions are the following:

Assumption 1: Boundedness ofg(t,0)g(t,0)g(t,0): There existsB> 0 such that ||g(t,0)||∞ ≤B.

Assumption 2: Semiglobal uniform boundedness of the partial derivatives ofg(t,x)g(t,x)g(t,x):

For an arbitrary δ> 0 there exist positive constants dgx(δ)> 0 and dgt(δ)> 0 independent

of time such that for all ||x(t)||∞ < δ, the partial derivatives of g(t,x)with respect to x and t

are piecewise continuous and bounded. We will consider g2(t,x(t)) since g1(t,x(t)) = 0.

We get:

∣∣∣∣∣∣∣∣∂g2(t,x)

∂x

∣∣∣∣∣∣∣∣∞

≤dg2x ,

∣∣∣∣∣∣∣∣∂g2(t,x)

∂t

∣∣∣∣∣∣∣∣∞

≤dg2t

Assumption 3: Partial knowledge of the system input gain: The system input gain

matrixω is unknown but considered to be nonsingular and row diagonally dominant. In

addition to that it has known bounds.

Assumption 4: Stability of the transmission zeros: The transmission zeros of the

transfer matrixHm(s) =C(sI−Am)−1Bm lie in the open left half plane.

The design parameters needed for the accomplishment of the proof of stability are the

feedback gain k and the transfer functionD(s) leading to the filterC(s) = ωkD(s)Im+ωkD(s)

.

The sufficient condition of stability is the following inequality:

||G(s)||L1 <ρr− ||kgC(s)H(s)||L1 ||r||L∞

−ρin

Lρrρr+B(5.27)

5.5. DESIGN OF A MULTI-VARIABLE CONTROLLER FOR DEPTH AND PITCH CONTROLIN UNDERWATER ROBOTICS 71

withG(s) =H(s)(I2−C(s)) andH(s) = (sI2−Am)−1Bm

ρr is defined as ρr > ρin where ρin = ||s(sI −Am)−1||L1ρ0 for a given ρ0 satisfying

||x0||∞ < ρ0 <∞.

Lρr =ρr(ρr)ρr

dfx(ρr(ρr)), ρr(ρr) = ρr+ γ1, with γ1 > 0 is an arbitrary positive constant.

Mathematical explanations concerning the L norms and other mathematical tools are

given in Appendix D.

5.5 Design of a multi-variable controller for depth and

pitch control in underwater robotics

Similarly to the control laws presented in the previous chapter and using the same no-

tation, we apply the proposed scheme to control depth and pitch of an underwater ve-

hicle. Let’s recall the equation of the studied vehicle dynamics (depth and pitch) in the

earth-fixed frame:

[M∗z 0

0 M∗ϑ

][z

ϑ

]+

[D∗z 0

0 D∗ϑ

][z

ϑ

]+

[−(W−B)

WzGcos(ϕ)sin(ϑ)

]=

[τ∗z +w

∗dz

τ∗ϑ+w∗dϑ

]

(5.28)

Combining (5.21) and (5.28), we get the dynamics of the system in the state space form:

[η1

η2

]=

[02×2 I2

02×2−D∗

rM∗r

][η1

η2

]−

[02×1

g∗rM∗r−w∗drM∗r

]+

[02×21M∗r

]ωτ∗r (5.29)

with η1 = [z ϑ]T and η2 = [z ϑ]T . In this case ω ∈ R2×2 is considered to be an iden-

tity matrix. Rewriting (5.29) in the form of (5.22) in terms of the state matrix Am and the

parametersω,θ and σ, we get:

[η1

η2

]=Am

[η1

η2

]+

[02×21M∗r

](ωua+θ(t)||η(t)||L∞

+σ(t)) (5.30)

y=

[1 0 0 0

0 1 0 0

][η1

η2

]=

[z

ϑ

](5.31)

where Am is obtained through a choice of km rendering the state matrix Hurwitz, with

Am ∈ R4×4 and Bm = [02×2,1M∗r]T ∈ R4×2. The parameters’ vector θ ∈ R2 includes the un-

certainties on the damping coefficients and is given by: θ= [∆(−D∗z ) , ∆(−D

∗ϑ)]

T . The pa-

rameter σ ∈R2 is a lumped parameter regrouping the gravitational and buoyancy forces as

well as the external disturbances σ=[−g∗z +w

∗dz,−g∗ϑ+w

∗dϑ

]T. The parameterω ∈ R2×2


is considered constant and will not be adapted for this case study as we have a precise

knowledge of the features of the thrusters of the system. The expression ||η(t)||L∞refers to

the infinity norm of the state vector η at instant t. As shown in equation (5.31), the outputs

of the system are the depth z and the pitch ϑ. The control input is computed in the earth

fixed frame and consequently should be mapped into the body-fixed frame. The system’s

control input is then computed as follows: u=K−1T−1JT (ua+um) ∈ R2, with ua and um

as explained above.

Remark 2: In the matricesAm andBm, the elementsM∗z ,M

∗ϑ andD∗

z ,D∗ϑ are likely to vary

since they depend on the orientation of the vehicle given that they are computed in the

earth-fixed frame. We have mentioned before thatAm and Bm should be constant and for

the sake of consistency, we replace these starred model elements withMz,Mϑ,Dz andDϑ.

This will guarantee forAm a constant desired dynamics. All the engendered uncertainties

due to this change will be compensated in the vectors of the controlled parameters θ and

σ that are to be adapted.

5.6 Conclusion

In this chapter, the nonlinear L1 adaptive controller has been presented. It is a novel

adaptive control scheme based on an architecture where robustness and adaptation are

decoupled. A background on this control architecture is presented. Its similarity with the

MRAC approach was given and a simple example was used to describe the various blocks

belonging to the architecture. Finally, the class of systems of interest was provided along

with the explanation of the sufficient condition to ensure the stability of the closed-loop

system. This adaptive scheme was designed to suit the control of depth and pitch in un-

derwater robotics.

CHAPTER

6Solution 3: A New Extension of L1

adaptive control

If everything seems under

control, you’re just not going fast

enough

MARIO ANDRETTI

Contents

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Limitation of the original L1 adaptive controller . . . . . . . . . . . . . . . . 74

6.3 Proposed extension of the L1 adaptive control . . . . . . . . . . . . . . . . . 75

6.4 Stability analysis of the extended L1 adaptive control . . . . . . . . . . . . 78

6.5 Design of a multi-variable controller for depth and pitch control in un-

derwater robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1 Introduction

The L1 adaptive control, explained in the previous chapter, was designed to be applied

on an underwater vehicle. This novel control scheme outperforms the nonlinear adaptive

state feedback one given its fast adaptation and robustness. Nevertheless, it holds some

limitations. A zero steady-state tracking error is guaranteed only for constant reference

trajectories. Indeed, similarly to Model Reference Adaptive Control (MRAC), the tracking

73

74 CHAPTER 6. SOLUTION 3: A NEW EXTENSION OF L1 ADAPTIVE CONTROL

error for time varying reference trajectories is only guaranteed to be bounded. Therefore, a

time lag can be noticed with the L1 controller due to the presence of a low pass filter in the

control loop. A very careful filter design should then be done to compromise between this

time lag and the desired performance bounds. To reduce the observed tracking error, we

propose a nonlinear proportional and a proportional integral derivative augmentation of

this controller. The architecture of the L1 controller will be therefore extended by adding

to the filtered input a nonlinear proportional (or proportional integral derivative) term and

then feeding the resultant to the prediction block and to the controlled system. The chap-

ter starts with the description of the modified architecture of the studied controller, then

a simulation example will be given to validate the effectiveness of the proposed solution.

A stability analysis is provided, and finally the extended controller will be designed for the

underwater’s vehicle control in depth and pitch.

6.2 Limitation of the original L1 adaptive controller

0 5 10 15 20 25 30 35 40 45 50−150

−100

−50

0

50

100

150

Time (s)

y(t)

Reference trajectorySystem output

Figure 6.1: Simulated example of the tracking performance of the originalL1 adaptive con-troller: the desired trajectory is displayed in dashed line and the output of the controlledsystem in solid line.

In order to illustrate the limitation of the originalL1 adaptive controller in terms of tra-

6.3. PROPOSED EXTENSION OF THE L1 ADAPTIVE CONTROL 75

jectory following, the example given in [Hovakimyan et Cao, 2010], page 29 is considered.

The open-loop dynamics of the system is expressed by the following state space equations:

x(t)=Ax(t)+b(u(t)+θTx(t)), x(0) = x0

y(t)= cx(t)(6.1)

withA=

[0 1

−1 −1.4

], b=

[0

1

], c=

[1 0

], θ=

[−4

−4.5

].

The proposed design parameters of the L1 controller are the following: the low pass fil-

terC(s) = 160s+160 and the adaptation gain is chosen to be Γ = 10000. The closed-loop sys-

tem is expected to track the following time varying reference trajectory: r= 100cos(0.2t).

The obtained simulation result is shown in Figure 6.1. We can clearly note the observed

time lag between the reference trajectory displayed in dashed line and the output of the

system displayed in solid line. This drawback can be a real problem when a precise track-

ing of a time varying reference trajectory is required. This could include for instance the

scanning of a dam by an underwater vehicle where all the joints should be verified. We

propose to solve this problem via the extended L1 adaptive controller explained in the

next section.

6.3 Proposed extension of the L1 adaptive control

The extended architecture consists in augmenting the original one with a nonlinear

P/PID feedback as illustrated in Figure 6.2. The extended section displayed in dotted lines

is able to reduce the time lag occurring in presence of a time varying reference trajectory

when the L1 adaptive controller (shown in dashed lines (in Figure 6.2) is employed. Two

solutions are then proposed here for this added block. A nonlinear proportional controller

or a proportional integral derivative controller can be used as the additional term to be

summed to the original filtered control input.

6.3.1 First variant: a PID based extension

For slow dynamical systems, a classical PID could be used to reduce the time lag previ-

ously described. The control input up(t) shown in Figure 6.2 is therefore expressed by:

up(t)=−KP e(t)−KI

∫ t

0

e(t) dt−KDde(t)

dt(6.2)

with KP, KI and KD are the proportional, integral and derivative gains respectively, and

e(t) the tracking error defined by:


Controlled System

Adaptation

State predictor

Control law with low pass filter

9

Output response y(t)

-

+

L1 Adaptive Controller

Nonlinear Proportional/

Propotional Integral Derivative

+

+Reference Trajectory r(t)

+

-

Extended block

Figure 6.2: Block diagram of the proposed extended L1 adaptive controller

e(t)=y(t)−r(t)with r(t) the reference trajectory andy(t) the measured output as shown

in Figure 6.2.

6.3.2 Second variant: a nonlinear proportional based extension

Conventional PID controllers involve constant gains multiplied by each of the forms of

the tracking error (proportional, derivative and integral). For a good trade-off between fast

response and reduced overshoot, a nonlinear PID could bring a performance improvement

when the controlled system has a relatively fast dynamics. In the previously introduced

illustrative example, we will refer to the use of a nonlinear proportional since it was enough

to reach the desired closed-loop performance in our case.

The added control input up(t) can then be expressed according to [Wang, 2012] by the

following:

up(t)=−g(e,α,δ) (6.3)

with

g(e,α,δ)=

a1|e|

αsgn(e) if |e|> δ

a2e

δ1−αif |e|≤ δ

where a1 and a2 are constant gains (for many applications, it might be preferable to

have a1 = a2 to avoid a discontinuity), e is the tracking error defined in section 3.1, α is a

design parameter with 0<α≤ 1 and δ is a threshold delimiting the transition between the

6.3. PROPOSED EXTENSION OF THE L1 ADAPTIVE CONTROL 77

low gains and the high gains. The idea behind this nonlinear proportional term is to have

small gains when the error is large and high gains when the error is small.

6.3.3 Validation in simulation on an illustrative example

0 5 10 15 20 25 30 35 40 45 50−150

−100

−50

0

50

100

150

Time (s)

y(t

)

Reference trajectory

Classical L1 controller

Extended L1 controller

Figure 6.3: Simulated performance of the L1 adaptive controller compared with the pro-posed extended L1 controller: the desired trajectory is displayed in red dashed line, the L1adaptive controller in blue solid line and the proposed extended controller in black solidline.

The proposed solution with the nonlinear proportional augmentation was applied to the

same previous example given in [Hovakimyan et Cao, 2010] page 29. In this section, we

compare the original L1 adaptive controller with the extended one. The gains and design

parameters are the same as those used for the previous simulation displayed in Figure 6.1.

The added proportional term considered for the extension of the original controller was

designed as follows:

up(t)=−g(e,r) (6.4)

with

g(e,r)=

25|e|0.1sgn(e) if |e|> 0.1r

100 e0.1r0.9

if |e|≤ 0.1r


The obtained simulation results are shown in Figure 6.3. The system output is dis-

played for both versions of the controller. The original L1 adaptive controller (dotted blue

line) exhibits a time lag that is eliminated when the proposed proportional extension is

added (solid black line). This example illustrates the benefits of extending the original ar-

chitecture proposed in [Hovakimyan et Cao, 2010].

6.4 Stability analysis of the extended L1 adaptive control

In this section, we analyse methods known from linear control theory to prove the ro-

bustness of the extended architecture namely: the Nyquist criterion and the stability mar-

gins. For this scope, a linear system is used (cf. Figure 6.4 for which the open-loop transfer

function is computed). This linearization is valid as long as the projection included in the

adaptation stage of the L1 controller is not required (i.e no saturation of the estimated

parameters). The Nyquist plot is displayed and the stability margins (phase and gain mar-

gins) are given for various design parameters. In our case, different values of the gains of

the PID extension and the adaptation gain will be tested in order to study their effects on

the robustness of the closed-loop system.

6.4.1 Illustrative example for the stability analysis

-

+

PID

+

+Reference Trajectory r(s)

+

-

+

+

+

-

++

Original Adaptive Controller

PID augmentation

indicates where the closed-loop is broken

Figure 6.4: Open-loop system with the proposed extended L1 adaptive controller for a lin-ear system.

In order to perform the stability analysis, let us now introduce the illustrative example

using the extended L1 controller and shown in Figure 6.4. It can be described as follows:

Controlled system:

x(t)=−x(t)+θ(t)+u(t) (6.5)

6.4. STABILITY ANALYSIS OF THE EXTENDED L1 ADAPTIVE CONTROL 79

with x(t) being the state of the system, θ the unknown disturbance and u(t) the control

input.

State predictor:˙x(t) =−x(t)+ θ(t)+u(t) (6.6)

where the hat symbol refers to the predicted state and estimated parameter.

Adaptation stage:˙θ(t) =−Γ x(t) (6.7)

with x= x−x is the estimate error and Γ is the adaptation gain. Given that this example

deals with a linear system. The adaptation law is taken to be proportional (i.e the error is

multiplied directly by the adaptation gain). Usually when implementing the L1 adaptive

controller a projection operator is used in the adaptation phase in order to ensure the

boundedness of the estimated parameters.

Control law:

u(t)=ua+uPID−C(s)(θ−r(t))+uPID (6.8)

with uPID=−KP e(t)−KI∫t0e(t) dt−KD

de(t)dt

and e(t) is the tracking error defined by: e(t)= x(t)−r(t)with r(t) the reference trajectory.

This example will now be used for stability analysis and comparisons.

6.4.2 Comparison between the original and the PID based extended L1

adaptive controller

The open-loop transfer function is computed in order to calculate the stability margins

of the augmented system. We break the block diagram of Figure 6.4 at the position of

the∫∫

symbol. We therefore get the resulting equation of the open-loop transfer function

Gextended, with negative feedback to be:

Gextended(s) =−(s+ Γ

s+1)uPID+ΓC(s)

s(s+1)+Γ(1−C(s))(6.9)

From the obtained open-loop transfer function, we can easily deduce the one of the nom-

inal controlled system without the extension by setting uPID to 0 and we get:

Gnominal(s) =ΓC(s)

s(s+1)+Γ(1−C(s))(6.10)

The Nyquist plot of both transfer functions is shown in Figure 6.5 and the computed

stability margins are summarized in the table 6.1 for the following design parameters:


Table 6.1: Comparison of the stability margins for both controllers

Nominal L1 Controller PID Extended L1 Controller

Gain margin 6 dB 7.9 dB

Phase margin 90 deg 99.8 deg

Γ = 100000,C(s)= 1s+1 . The PID parameters were set to: KP = 3,KI= 0.5, and KD= 0.2.

Both Nyquist diagrams never encircle the critical point (−1,0). Since the number of

anti-clockwise encirclements is equal to the number of unstable poles of the open-loop

transfer function, we deduce that both closed-loop systems are stable. However, it is worth

to note that the stability margins are slightly increased for the extended controller as seen

in Table 6.1. It has to be reminded that the desired effect of the PID extension is to reduce

the time-lags induced in presence of a time varying reference trajectory without affecting

the overall closed-loop stability of the system. Therefore it would be important to keep the

stability margins unchanged or slightly improved.

6.4.3 Effects of the PID feedback gains on the stability

The extended L1 controller was proven to satisfy the Nyquist criterion of stability for

chosen specific gains of the PID. In this section, we prove in this section that a wide vari-

ety of these gains enable preserving the closed-loop stability. Starting from the parameters

used in the previous section, each feedback gain of the PID is varied alone and the cor-

responding stability margins are computed for each case (cf. Tables 6.2, 6.3, and 6.4). In

addition, the corresponding Nyquist plots are also depicted in Figures 6.6, 6.7, and 6.8. We

notice that augmenting the proportional gain has a very small effect on the stability mar-

gins given that the values in Table 6.2 nearly remain constant even after multiplying the

proportional gain by 10. The integral gain (cf. Table 6.3) does not affect the gain margin

(GM) but it reduces the phase margin (PM), whereas the derivative gain (cf. Table 6.4) has

a big impact on the values of the stability margins since a small increase in this parameter

value increases both phase and gain margins. In conclusion, only the integral coefficient

is prone to affect significantly the stability of the system and therefore should be chosen

carefully. However, for the range of the above tested values, the phase and gain margins

remain large enough to ensure the stability of the system.


Real Axis

Ima

gin

ary

Axi

s

−1.5 −1 −0.5 0 0.5 1 1.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

Classical L1 controller

PID Extended L1 controller

Figure 6.5: Nyquist plot of the system (6.10) corresponding to the case of the original L1controller (solide blue line) and (6.9) corresponding to the PID based extended one (dottedred line).

Table 6.2: Effects of changing the proportional gain on the stability margins.

Parameter Value Parameter Value Parameter Value

KP 3 KP 15 KP 30

KI 0.5 KI 0.5 KI 0.5

KD 0.2 KD 0.2 KD 0.2

PM 100 deg PM 101 deg PM 101 deg

GM 7.9 dB GM 8 dB GM 8.1 dB

6.4.4 Effects of the adaptation gain on the stability

The adaptation gain was also varied to see its effect when a PID based extended con-

troller is used with the following chosen PID gains: KP = 15,KI = 2.5, and KD = 0.2. We


-1.5 -1 -0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

Real Axis

Ima

gin

ary

Axi

s

Figure 6.6: Nyquist plots of the open-loopsystem for the case of the PID based ex-tended L1 controller for different values ofthe proportional gain: Kp = 3 in solid blueline,Kp= 15 in dashed green line, andKp=30 in red dotted line.

-1.5 -1 -0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

Real Axis

Ima

gin

ary

Axi

s

Figure 6.7: Nyquist plots of the open-loopsystem for the case of the PID based ex-tended L1 for different values of the inte-gral gain: KI = 0.5 in solid blue line, KI =2.5 in dashed green line, and KI = 5 in reddotted line.

-1.5 -1 -0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

Real Axis

Ima

gin

ary

Axi

s

Figure 6.8: Nyquist plots of the open-loopsystem for the case of the PID based ex-tended L1 controller for different values ofthe derivative gain: KD = 0.1 in solid blueline, KD = 0.2 in dashed green line, andKD= 0.3 in red dotted line.


Table 6.3: Effects of changing the integral gain on the stability margins.


KP 3 KP 3 KP 3

KI 0.5 KI 2.5 KI 5

KD 0.2 KD 0.2 KD 0.2


GM 7.9 dB GM 7.9 dB GM 7.9 dB

Table 6.4: Effects of changing the derivative gain on the stability margins.


KP 3 KP 3 KP 3

KI 0.5 KI 0.5 KI 0.5

KD 0.1 KD 0.2 KD 0.3


GM 6.9 dB GM 7.9 dB GM 9.1 dB

notice, from the Nyquist plot in Figure 6.9 and the Table 6.5, that the phase margins re-

main unchanged, while a slight decrease in the gain margin is observed which indicates

that similarly to the original controller the adaptation gain does not alter the stability of

the system for the extended L1. We can therefore conclude that this new proposed con-

troller also guarantees the property of the decoupling between robustness and adaptation.

Table 6.5: Effects of changing the adaptation gain on the stability margins.

Adaptation Gain Gain Margin Phase Margin

5000 8.9 dB 101 deg

10000 8.4 dB 101 deg

100000 8 dB 101 deg


Real Axis

Ima

gin

ary

Axi

s

−1.5 −1 −0.5 0 0.5 1 1.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

Figure 6.9: Nyquist plot of the system given in (6.9) for different values of the adaptationgain : Γ = 5000 (solid blue line), Γ = 10000, (dashed green line) and Γ = 100000 (dotted redline.)

6.5 Design of a multi-variable controller for depth and

pitch control in underwater robotics

We consider the control of depth and pitch for an underwater vehicle. Let us recall the

equations of the dynamics to be controlled.[η1

η2

]=Am

[η1

η2

]+

[02×21M∗r

](ωua+θ(t)||η(t)||L∞

+σ(t)) (6.11)

y=

[1 0 0 0

0 1 0 0

][η1

η2

]=

[z

ϑ

](6.12)

where Am is the Hurwitz state matrix denoting the desired dynamics of the system, with

Am ∈ R4×4 and Bm = [02×2,1M∗r]T ∈ R4×2. As mentioned in the previous chapter, the sub-

script r refers to the reduced model describing the depth and pitch dynamics. The pa-

rameters’ vector θ ∈ R2 includes the uncertainties on the damping coefficients and is

6.6. CONCLUSION 85

given by: θ = [∆(−D∗z ) , ∆(−D

∗ϑ)]

T . The parameter σ ∈ R2 is a lumped parameter re-

grouping the gravitational and buoyancy forces as well as the external disturbances σ =[−g∗z +w

∗dz,−g∗ϑ+w

∗dϑ

]T. The parameter ω ∈ R2×2 is considered constant and will not

be adapted for this case study as we have a precise knowledge of the thrusters’ features.

The expression ||η(t)||L∞refers to the infinity norm of the state vector η at instant t. A de-

tailed description of this model is found in section 5.5. As shown in equation (6.12), the

outputs of the system are the depth z and the pitch ϑ. The control input is computed in

the earth-fixed frame and consequently should be mapped into the body-fixed frame. The

controller described in chapter 5 is compared with the extended one described in Figure

6.2.

The control input in this case is expressed by: u= K−1T−1JT (ua+um+up) with ua and

um explained in the previous chapter and up given by (6.2).

Remark 1: The model parameters M∗z ,M

∗ϑ and D∗

z ,D∗ϑ are present in the matrices Am

and Bm as explained previously in section 5.5. The parameters are likely to vary since they

depend on the orientation of the vehicle given that they are computed in the earth-fixed

frame. We have also mentioned before that Am and Bm should be known and constant

since they delimit the dynamics of the system. For the sake of consistency, we replace

these starred model elements with the constant onesMz,Mϑ,Dz andDϑ expressed in the

body-fixed frame. This will guarantee for Am a constant desired dynamics. The uncer-

tainties caused by this modification will be compensated in the vectors of the controlled

parameters θ and σ that are to be adapted.

6.6 Conclusion

In this chapter, a new time varying extension of the L1 adaptive controller is proposed.

The original architecture of this control scheme was modified to include a PID augmenta-

tion term in order to reduce the tracking errors in presence of a new time varying reference

trajectory. The modified architecture was first tested in simulation on a simple illustrative

example borrowed from the literature. The stability analysis of the controlled system with

the proposed extended version of the controller is presented. It shows clearly that the new

control architecture guarantees also the stability margins. Indeed, Nyquist plots and values

of the phase and gain margins were provided for several gains of the PID as well as for sev-

eral values of the adaptation gain. It was therefore proven that the stability margins were

conserved for a big range of design parameters. Moreover, to completely eliminate the

tracking error, another extension has been proposed which relies on a nonlinear augmen-

tation. In fact, this latter augmentation allows the presented system to follow a sinusoidal

trajectory without any time lags.

Part III

Experimental Results

87

CHAPTER

7Experimental case study: the AC-ROV

underwater vehicle

Our goal is not to build a

platform; it’s to be cross all of

them.

MARK ZUCKERBERG

Contents

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2 General features of the AC-ROV vehicle . . . . . . . . . . . . . . . . . . . . . 90

7.3 Thrusters’ configuration and characteristics . . . . . . . . . . . . . . . . . . 91

7.4 Hardware architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.1 Introduction

The vehicle used for experimental validation of the proposed control schemes is a mod-

ified AC-ROV. The original AC-ROV vehicle is manufactured by AC-CESS (http://www.ac-

cess.com), a Scottish manufacturer for ROVs. The AC-ROV, in its commercial configuration,

is equipped with a depth sensor and a video camera, and it is controlled by a 3D-mouse.

We added an Inertial Measurement Unit (IMU), and we brought modifications allowing

the control of each motor’s speed via a control PC. This chapter describes the underwater

vehicle, the hardware architecture, and the thrusters along with their configuration and

characteristics.

89

90 CHAPTER 7. EXPERIMENTAL CASE STUDY: THE AC-ROV UNDERWATER VEHICLE

7.2 General features of the AC-ROV vehicle

(a) AC-ROV(b) AC-ROV commercial con-figuration

Figure 7.1: View of the AC-ROV vehicle (a) and its commercial setup (b)

Table 7.1: Main characteristics of the AC-ROV

Size (cm) 20.3×15.2×14.6

Weight (Kg) 3

Depth (m) 75

Thrusters 6

DOF 5

The AC-ROV underwater vehicle (cf. Figure 7.1(a)) weighs 3 kg in the air and has a

rectangular shape with 20.3 cm height, 15.2 cm length and 14.6 cm width. It can dive up

to 75m. This vehicle has a neutrally buoyant tether, the diameter of which is 9.3mm. It has

6 thrusters controlling 5 degrees of freedom (all degrees of freedom, except roll). It is also

equipped with a frontal video camera. The operator controls the vehicle with a "space-

mouse" (3D-mouse), and can read the depth displayed on the video monitor. Table 7.1

summarizes the main characteristics of the AC-ROV in its commercial configuration.

7.3. THRUSTERS’ CONFIGURATION AND CHARACTERISTICS 91

7.3 Thrusters’ configuration and characteristics

7.3.1 Thrusters’ configuration

The propulsion system of the AC-ROV underwater system consists of six thrusters con-

trolling five degrees of freedom. These actuators allow controlling the vehicle’s orientation

in pitch and yaw, as well as all translational motions along the x,y, and z axes. The yaw

control is provided thanks to the differential speed control of the thrusters 1,2,3, and 4

exerting the forces f1,f2,f3, and f4 as depicted in Figure 7.2(a). These four thrusters also

control the translations along x and y axes. Depth and pitch control are both obtained

using thrusters 5 and 6 exerting the forces f5 and f6, whereas the roll is left uncontrolled.

However, the roll dynamics remains stable due to the damping parameter Dxx (first ele-

ment in the damping matrix defined in section 3.3), and due to the metacentric distance.

(a) (b)

Figure 7.2: View of the vehicle under study with the orientation of the thrusts (a) as well asa transversal cut showing the positions of the propellers and motors (b).

The forces and torques generated by the thrusters are considered as the control inputs

affecting the degrees of freedom of the vehicle. Taking in consideration the configuration

of the motors as displayed in Figure 7.3 and the blade pitches, we can express the control

vector acting on the vehicle according to the following:


1

2

3

4

5

6

3

Figure 7.3: Transversal cut of the underwater vehicle and its body-fixed frame (xbybzb).Rotational directions of the propellers are depicted by yellow arrows. The blue propellershave a positive blade pitch, (i.e right hand rotation) and the yellow ones have a negativeblade pitch (i.e left hand rotation).

τ=K

Bp1u1cos(π4 )−Bp2u2cos(

π4 )+Bp3u3cos(

π4 )−Bp4u4cos(

π4 )

Bp1u1sin(π4 )+Bp2u2sin(

π4 )−Bp3u3sin(

π4 )−Bp4u4sin(

π4 )

−Bp5u5−Bp6u6

0

L1(Bp5u5−Bp6u6)

L2sin(π4 )(Bp1u1−Bp2u2−Bp3u3+Bp4u4)

(7.1)

whereK is the force coefficient in Newton.Volt−1 and has been identified after several tests

performed on the motors. τ= [τxτyτzτϑτϕτψ]T ∈R6 is the vector of control inputs depict-

ing for every degree of freedom the forces and moments (in Newton and Newton.meter).

The control inputs of the motors are denoted ui, with i being the number of the consid-

ered thruster from 1 to 6. L1 is the distance between thrusters 5 and 6 and the origin of

the robot’s frame. L2 is the distance between thrusters 1,2,3, and 4 and the origin as well.

Bpi is the blade pitch of the propellers with i being the number of the thruster from 1 to

6. Its value is either 1 if the blade pitch is positive (the thrust generated by the propeller

respects the right hand rule of the propeller’s rotation) or −1 if the blade pitch is negative

(the thrust generated by the propeller is opposite to the right hand rule of the propeller’s

7.3. THRUSTERS’ CONFIGURATION AND CHARACTERISTICS 93

rotation). According to Figure 7.3, the propellers with a positive pitch are displayed in blue

and the ones with a negative one are displayed in yellow. The vector Bp is therefore given

by the following:

Bp=

Bp1Bp2Bp3Bp4Bp5Bp6

=

1

1

−1

−1

−1

1

(7.2)

For control purposes, it is necessary to be able to send a voltage to each motor to

achieve the desired force or moment. Equation (7.1) can be rewritten in terms of a thruster

configuration matrix T and the vector of motors’ input voltageu as:

τ=KTu (7.3)

where T ∈ R6×6 is the thrusters’ configuration matrix taking into account the position

and orientation of the thrusters, thus allowing to determine the associated forces in the

body frame. By writing explicitly the above expression and incorporating the blade pitch

in matrix T, we get:

τ=K

cos(π4 ) −cos(π4 ) −cos(π4 ) cos(π4 ) 0 0

sin(π4 ) sin(π4 ) sin(π4 ) sin(π4 ) 0 0

0 0 0 0 1 −1

0 0 0 0 0 0

0 0 0 0 −L1 −L1

L2sin(π4 ) −L2sin(

π4 ) L2sin(

π4 ) −L2sin(

π4 ) 0 0

u1

u2

u3

u4

u5

u6

(7.4)

7.3.2 Thrusters’ characteristics

The thrusters’ characteristics are displayed on Figure 7.4(a). We can note the presence

of a dead zone and a strong hysteresis. Moreover, the friction and the high non-linearities

also induce the absence of repeatability. In fact, not only the non-linear zone is different

for each motor but it can also vary from one experiment to the other. To overcome this

problem, an additional buoyancy has been added on top of the AC-ROV as displayed in

Figure 7.4(b). This implies that the vehicle needs to permanently compensate the induced

lift force and therefore compels the thrusters to run in their linear zone. In fact, the new

floatability (B−W)0 (where B is the buoyancy andW the weight of the vehicle) is chosen


Linear zone

Hysteresis

Nominal

operating point

Linear zone

Hysteresis

(a) (b)

Figure 7.4: Thrusters’ charactersitics (a) and added buoy on top of the AC-ROV to remedyfor the hysteresis and nonlinearities (b).

to be in the middle of the linear part of the thrusters’ characteristic curve as depicted on

Figure 7.4(a). Having the thrusters operating in their linear zone, we can now deduce the

coefficient K of (7.4) by calculating the slope of the line located in the linear zone of Figure

7.4(a). The offset (−F0) is suppressed in the software, thus allowing to keep the propor-

tionality described by equation (7.3). It is important to note that this solution works for

our case since only depth and pitch are to be controlled. The two thrusters in use are the

vertical ones and the only ones affected by this parameter change.

Obviously, this technique involves the loss of three quarters of the thrusting capabili-

ties, which is not satisfying. However, this allows us to validate the control schemes pro-

posed in spite of the very poor performances of the AC-ROV ’s thrusters. With higher quality

thrusters (such as Seabotix BTD150), these undesired effects could be neglected or com-

pensated by software, thus allowing to use the full range of thrust.

7.4. HARDWARE ARCHITECTURE 95

Figure 7.5: Schematic view of the hardware architecture of the AC-ROV prototype.

7.4 Hardware architecture

The hardware architecture of the commercial AC-ROV displayed on Figure 7.1(b) has

been deeply modified in order to become a useful experimental platform. A digital pres-

sure sensor has been added for depth measurement, and a 6-DOF IMU has also been in-

corporated to measure roll, pitch, and yaw, along with their respective rotational veloci-

ties. The IMU as well as the pressure sensor are connected to the control PC through a

specific electronic board based on a PIC 16F1825 microcontroller (Microchip). This 8-bit

microcontroller receives samples from both sensors at a 100ms samping rate, and merges


the data into a unique packet that is sent to the PC through the tether. This transferred

data can then be accessed by the control software though a RS485 USB port adapter (VS-

com USB-2COMiSI-M). Once the control law has been computed by the control PC, the

control inputs are transmitted through the ethernet link to the power stage. A rabbit core

RCM3900 module converts the ethernet inputs into SPI packets, allowing to control each

of the 6 motor drive boards. Each motor drive delivers a Pulse Width Modulation (PWM)

output directly to each ROV’s motor through the tether at very low frequency (≈ 50Hz).

Figure 7.5 shows a schematic view summarizing the various components of the vehicle’s

hardware and their interactions, and Figure 7.6 gives an overview of the whole used ex-

perimental testbed. It includes the underwater vehicle, the control PC and the hardware

case.

1©ւ

2©

↑3©

↑

4©

↓

5©

↑

6©

↓7©

↑

8©

↑

9©

↑

Figure 7.6: View of the AC-ROV experimental testbed: 1© Control PC, 2© Power input, 3©

Emergency stop button, 4© Video in, 5© Tether plug, 6© Ethernet plug, 7© Video Capture,8© Tether, 9© AC-ROV.

7.5 Conclusion

In this chapter the experimental testbed was presented. The underwater vehicle un-

der study is the AC-ROV an industrial robot, the hardware of which has been modified,

yielding the platform used for the validation of the proposed control schemes. A general

description of the main features of the ROV was presented. The thrusters’ configuration

and characteristics were provided along with the hardware architecture.

CHAPTER

8Experimental results of the proposed

control schemes

It is the weight, not numbers of

experiments that is to be

regarded.

ISAAC NEWTON

Contents

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.2 Description of the investigated experimental scenarios . . . . . . . . . . . 98

8.3 Application of the PID controller . . . . . . . . . . . . . . . . . . . . . . . . . 99

8.4 Application of the NASF controller . . . . . . . . . . . . . . . . . . . . . . . . 103

8.5 Application of the L1 adaptive controller . . . . . . . . . . . . . . . . . . . . 109

8.6 Application of the extended L1 adaptive controller . . . . . . . . . . . . . . 114

8.7 Comparison among the various proposed controllers . . . . . . . . . . . . 118

8.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.1 Introduction

To validate the proposed control schemes presented in chapters 4, 5 and 6, real-time ex-

periments have been performed on the AC-ROV underwater vehicle described in chapter

7. As mentionned above, two degrees of freedom have been considered to be controlled,

namely depth and pitch. Time varying reference trajectories have been designed to be

97

98 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES

tracked on these two positions for different experimental scenarios. The aim behind these

tests is to observe the behavior of the closed-loop system in different operating conditions

and compare the performance of the controlled underwater vehicle with the different pro-

posed controllers.

8.2 Description of the investigated experimental scenarios

The real-time experiments have been performed in a 10 m3 pool (4 m wide hexagon

and 1.2 m depth). The tether has been sufficiently deployed to avoid inducing additional

drag into the dynamics of the vehicle. For all the experiments, the considered initial posi-

tion of the vehicle is on the surface (horizontal static position) then it is controlled to follow

time varying trajectories in depth and pitch. Three variations for each degree of freedom

are considered. The variations for the depth are the following: at t= 0 s, the depth varies

from 0 m to 0.4 m in 15 seconds, then at t= 45 s, from 0.4 m to 0.6 m in 15 seconds, and

finally at t= 100 s, from 0.6m to 0.5m in 15 seconds. As for the pitch angle, the trajectory

goes at t = 20 s, from 0 deg to 5 deg in 5 seconds, then at t = 55 s, from 5 deg to 0 deg

in 5 seconds, and finally at t= 100 s, from 0 deg to −5 deg in 15 seconds. Each degree of

freedom remains In fact, the tested trajectories (in pitch and depth) have been chosen so

as to minimize the risk of saturation of the thrusters (i.e a maximum force of 1.1N for each

thruster). That is why the dive of the robot follows a staircase pattern and the pitch is lim-

ited to ±5 deg. It is worth to note that these results can easily be extended to larger scales

and faster trajectories in case of an underwater vehicle equipped with more powerful actu-

ators and a deeper pool. A preliminary remark should be made concerning the pitch angle:

in fact, regardless of the used controller, the pitch angle shows some oscillations during the

first 10 seconds. This behavior is due to the fact that the vehicle starts its trajectory from

the surface, thus leading to disturbances that cannot be compensated since the propellers

are not fully immersed during this initial period of the experiment.

(a) (b)

Piece of polyester foamfixed on top of the robot

(c)

Figure 8.1: View of the AC-ROV in different operating conditions: (a): Nominal case, (b):Buoyancy change, (c): External disturbance (Mechanical Impact).

8.3. APPLICATION OF THE PID CONTROLLER 99

In order to test the proposed control schemes in different operating conditions, three

experimental scenarios displayed in Figure 8.1 were performed for each controller ,

namely:

i) Scenario 1: Control in nominal conditions

The objective of this scenario is to control the depth of the AC-ROV without consider-

ing any external disturbance. The feedback gains for each controller have been tuned

to accommodate this case and were kept unchanged for the rest of the experiments.

ii) Scenario 2: Robustness towards parameter uncertainty

The model of the vehicle was changed right before starting this scenario by the addi-

tion of a rectangular piece of polyester foam (cf. Figure 8.1). Consequently, a change of

buoyancy of +0.2 N was brought to the system bringing a variation of approximately

15% with respect to the nominal value of the parameter (W−B). Such a variation cor-

responds in real conditions for instance to the situation where the vehicle encounters

a sudden change in the water’s salinity.

iii) Scenario 3: Punctual external disturbance rejection

In this scenario, while following the same trajectories as the previous scenarios, the

underwater vehicle is submitted to several isolated vertical mechanical impacts push-

ing it downwards. The objective of this experiment is to see whether the proposed

controllers are able to steer the system back to its regulated position. Such a situation

occurs for instance where the vehicle hits a rock or an underwater structure, or collides

with another vehicle or a floating obstacle.

In summary, the aim behind these three scenarios is to evaluate the performance of the

closed-loop system in different operating conditions for each of the proposed controllers.

For this reason, the feedback gains of each controller have been tuned for the nominal

case and kept unchanged for the rest of the experimental scenarios despite the eventual

changes in the model or its environment. A comparison of the four proposed controllers

will be given at the end of this chapter in terms of trajectory following, robustness level and

some other closed-loop performance characteristics.

8.3 Application of the PID controller

8.3.1 Controller’s parameters tuning

As seen in chapter 4, two tuning methods were used in order to find the feedback gains

of the PID controller. The parameter setting used for the integral plus dead time (IPDT)

model was applied only for the depth and the gains found were the following:


Kpz = 4.89, Kiz = 0.46, Kdz = 13.92

Concerning the pitch angle, the Ziegler Nichols method was used and the following

gains were obtained:

Kpϑ = 0.1, Kiϑ = 0.3, Kdϑ = 0.05

8.3.2 Real-time experimental results

For each experimental scenario, the evolution of the controlled positions as well as the

control inputs, generated by thrusters 5 and 6 are displayed. The main advantage of the

PID control scheme relies in the simplicity of its implementation and the satisfactory re-

sults it can achieve. That is why it is commonly used in most of the industrial applica-

tions. The performances of this controller rely on the combination of three actions: the

proportional, the integral and the derivative. The experiments below will put upfront its

advantages and drawbacks.

Scenario 1: Control in Nominal Conditions

Figure 8.2 displays the evolution of the vehicle’s depth and pitch, as well as the con-

trol inputs. The PID controller needs around 34 seconds to reach the depth of 40 cm and

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch

(a) Time history of the measured depthposition z and pitch angle ϑ as well as theirrespective desired trajectories.

0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6

(b) Time history of the forces generated bythe two thrusters controlling z and ϑ.

Figure 8.2: Application of the PID controller (Control in nominal case): (a) evolution ofthe system outputs’ responses (z and ϑ) and (b) evolution of the control inputs.

8.3. APPLICATION OF THE PID CONTROLLER 101

17 seconds to reach the steady state position at 60 cm (cf. Figure 8.2(a)). As for the last

variation in depth and pitch angle, a good tracking of the desired trajectory is observed.

Note that the initial depth delay is caused by the time needed for the integral part to com-

pensate the flottability of the robot ((W−B) ≈−1.2 N) but mainly it is caused by the sat-

uration of thruster 6 the first 50 seconds (clearly visible in Figure 8.2(b)and occuring at

1.1 N). This saturation is due to the steep slope in depth coupled with the simultaneous

variation in pitch. We do not observe this phenomenon for the last variation of depth and

pitch occurring at t= 100 s since the slope for the depth is smoother (a variation of 10 cm

in 15 seconds, compared to the initial one of 40 cm in 15 seconds). What can also be

observed from Figure 8.2(b), is that the control input is very noisy with a maximal noise

amplitude of 0.4 N.

Scenario 2: Robustness towards Modeling Uncertainties: Change in buoyancy

The additional buoyancy (illustrated in Figure 8.1(b)) incorporated into the system dis-

turbs in a persistent way the motion of the vehicle that would tend to float more. The

controlled system for this scenario responded with a delay of 5 seconds for the first step in

depth compared to the nominal case as seen in Figure 8.3(a). The remaining tracking in

depth is satisfactory except for the presence of maximal overshoots of 2.3 cm during the

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.3: Application of the PID controller (Robustness towards parameter uncer-

tainty): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution of thecontrol inputs.


b!]

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150

−20

−10

0

10

20

30

40

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.4: Application of the PID controller (Punctual external disturbance rejection):

(a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution of the controlinputs.

leveling at 60 cm and 50 cm. In the evolution of the pitch angle, we observe oscillations

of higher amplitude and frequency all along the trajectory. The tracking performance was

not affected but we note that these oscillations are more important when the first varia-

tion in pitch occurs. Similarly to the nominal case for this controller, we relate the delay

in the convergence of the depth to the clear saturation of the thrusters (at F= 1.1 N) and

mainly the thruster 6 (cf. Figure 8.3(b)). This saturation can also be behind the presence

of high amplitude oscillations in pitch. In fact, when an additional buoyancy was added, a

larger force to be generated by the thrusters is required in order to immerse the vehicle. We

also observe that when the pitch angle is non zero, the force required from each thruster is

not the same. This uneven force distribution tends to saturate one thruster over the other.

The total required force increased from −1.2 N to −1.4 N which is less than the maximal

admissible one (2.2 N by summing the maximal force of both thrusters).

Scenario 3: External Disturbance Rejection: Mechanical Impact

As specified above, an external punctual disturbance has been applied on the vehicle

through a mechanical impact pushing the robot downwards. This disturbance was applied

several times along the trajectories at different amplitudes. In Figure 8.4(a), we observe

that when the external disturbance occurs at steady state (time t=70 s), the settling time

of the depth is around 10 s and about 5 s for the pitch. The former degree of freedom had

8.4. APPLICATION OF THE NASF CONTROLLER 103

an overshoot of approximately 10 cm for a disturbance variation of 5 cm whereas the latter

one oscillated up to 20 deg for a variation of 6 deg. A disturbance was also made during

a variation in the reference trajectory (time t=105 s) and the recovery times for both con-

trolled coordinates were the same. When the vehicle is subjected to a bigger disturbance,

the overshoot reaches its maximum (and the vehicle sufaces) resulting in a settling time of

40 s for the depth and 10 s for the pitch. As it has already been seen in the previous two

scenarios, a saturation of the thrusters is observed which worsens the closed-loop behav-

ior of the system. It is easy to correlate in Figure 8.4 the larger overshoots of the positions

with the saturation of the thrusters.

8.4 Application of the NASF controller

8.4.1 Controllers’ parameters

All the parameters and feedback gains used in these experiments are summarized in Ta-

ble 8.1. The vector of the estimated parameters has been initialized with our rough initial

knowledge about the system. The better the initialization is, the faster the controller will

adapt the parameters and the better the closed-loop performance will be. Moreover, bad

initialization could even lead to instability of the closed-loop system. The mass Mz was

easily obtained since it is the mass of the robot and the mass parameterMϑ was approx-

imated by calculating the inertia under the assumption that the robot is a homogeneous

cube. The damping was approximated from free diving trials (by measuring the in water

fall velocity loaded with a known mass) and the floatability (W−B) was deduced from the

experiments. In fact, (W−B) can be defined as the force needed to keep the robot at a

static position underwater. As for the parameter (zGW), it was estimated close to 0 given

that the distance between the center of mass and the center of buoyancy is very small. The

remaining terms were calculated under the condition of satisfying the stability criterion

described in Appendix B. Concerning the adaptation gain, the higher it is, the faster the

closed-loop response will be at the price of deteriorating the stability and the robustness

of the closed-loop system. In this scheme, the acceleration in the earth frame is linearized

and PID gains are used. These latters should also be tuned in order to ensure that the error

goes to zero.


In the following, for each experimental scenario, the evolution of depth and pitch out-

puts as well as the control inputs, generated by thrusters 5 and 6 are displayed. This

scheme is expected to bring improvements over the PID controller in terms of tracking


Table 8.1: Parameters’ values of the NASF controller used in the experiments.

Parameter Description Value Parameter Description Valueco Constant gain 0.3 Kpz Proportional gain of z 1

c1 Constant gain 0.1 Kiz Integral gain of z 0.042

Mzinitial Initial value of the mass 3 kg Kdz Derivative gain of z 1

Dzinitial Initial value of the damping along z axis 0.2N.s/m Kpϑ Proportional gain of ϑ 0.4

Mϑinitial Initial value of the mass of ϑ 0.1kg.m Kiϑ Integral gain of ϑ 2

Dϑinitial Initial value of the rotational damping along y axis 0.01N.s/m Kdϑ Derivative gain of ϑ 0.6

(W−B)initial Initial value of the floatability −1.2 N Γz Adaptation gain of z 1

zGWinitial Initial value of the restoring torque of ϑ 0.05N.m Γϑ Adaptation gain of ϑ 1

performance, robustness and energy consumption. The main drawbacks seen with the

previous control scheme was the saturation of the thrusters that degrades the closed-loop

performance of the system and the time delay observed for the convergence to the first

variation in depth. For the following experimental results, we only estimate the param-

eters pertaining to gravity and buoyancy since they are the ones with the biggest impact

on the dynamic model. In fact, parameter excitation is needed in order to induce changes

for the estimation of the model parameters. The suggested trajectory excites mainly the

parameters (W−B) and (zGW) which have a bigger effect on the dynamics of the vehicle.

The other parameters did not vary enough and therefore are not displayed.


Figure 8.5 displays the evolution of the vehicle’s position (in terms of depth and pitch)

as well as the control inputs. According to the obtained results, a fast convergence to the

initial depth variation and a good trajectory tracking can be observed. In fact, the system

converges to a depth of 40 cm in 25 seconds and this convergence time also coincides with

the time needed for the parameters (W−B) and (zGW) to converge. The evolution of these

parameters are depicted in Figure 8.6, they reach a steady state value of −1.2N for (W−B)

and −0.05 N.m for (zGW). A small time lag is generally observed in the tracking for both

degrees of freedom when the trajectory is varying. Oscillations and a clear time lag can

be observed on the measured pitch for the last variation in this degree of freedom. This is

due to the simultaneous variation in both degrees of freedom. The control inputs depicted

in Figure 8.5(b) show a combined force of ≈ 1.2 N needed to keep the robot floating and

compensate the floatability. This value indeed coincides with the estimated one. The os-


0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.5: Application of the NASF controller (Control in nominal case): (a) evolution ofthe system outputs’ responses (z and ϑ) and (b) evolution of the control inputs.

−1.5

−1

−0.5

0

W-B

(N)

0 50 100 150−0.5

−0.4

−0.3

−0.2

−0.1

0

Time (s)

G

z W

(N

.m)

Figure 8.6: Application of the NASF controller (Control in nominal case): Time history of

the estimated parameters gr= [W−B, zGW]T .

cillations in the control input have a maximum amplitude of 0.1 N and no saturations of

the thrusters are observed.


Depth and pitch obtained with the NASF controller under this scenario are depicted in


Figure 8.7(a), where we observe an additional delay for the convergence of the first depth

variation. In fact, the slope of diving of the vehicle remained almost the same as the nom-

inal case but the desired depth was reached within about 34 seconds. This was caused by

the time needed for the parameters to converge to their new values (cf. Figure 8.8). It is

worth to note that adaptive controllers do not necessarily ensure the convergence of the

estimated parameters to their true values [Slotine et Weiping, 1991] in order to obtain the

convergence of the system’s output to its desired position. The control law detailed in sec-

tion 4.3 ensures the boundedness of the parameters but not necessarily their convergence

to the real values. However, it is worth to mention that the value of (W−B) converges

to 1.4 N being the new floatability of the system. We observe for the depth a steady state

error of 1.5 cm when the vehicle levels at 60 cm. Finally for this same degree of freedom,

we can also observe a small increased time lag between the reference trajectory of the last

variation in depth and the measured position. Concerning the pitch, the same tracking

performance observed in the nominal case is preserved. Concerning the control inputs

generated by this controller, and depicted in Figure 8.7(b), we can observe that the robot’s

thrusters are exerting more effort and they maintain a sum of approximately 1.4 N. The

same remark made for the previous case concerning the power consumption holds for this

scenario: the profiles of the thrusters in the case of the NASF are smooth for this case and

most importantly, no saturation was observed.

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.7: Application of the NASF controller (Robustness towards parameter uncer-

tainty): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution of thecontrol inputs.


−1.5

−1

−0.5

0

0 50 100 150−0.5

−0.4

−0.3

−0.2

−0.1

0

Time (s)

W

-B(N

)Gz W

(N

.m)

Figure 8.8: Application of the NASF controller (Robustness towards parameter uncer-

tainty): Time history of the evolution of the parameters gr= [W−B, zGW]T .


The evolution of the measured positions under the NASF controller are displayed in Fig-

ure 8.9(a). The ability of rejecting external disturbances can be considered as satisfactory.

The mechanical impact occuring at time t=150 s was recovered within 8 s without any

overshoot. However, for the depth when the disturbance occurs right before the system

reaches steady state (after a disturbance or a variation in the trajectory, ex: time t=28 s and

t=122 s), an overshoot of 15 cm is noted and the settling time is 10 s. As for the pitch angle,

the maximal overshoot was of 15 deg and the settling time is around 4 s. The control in-

puts displayed in Figure 8.9(b) never saturate which explains the satisfactory closed-loop

behavior of the vehicle under the NASF controller. It is also important to mention that the

variation of the parameters upon each mechanical impact as displayed in Figure 8.10 helps

in stabilizing the system.


0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−15

−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)C

on

tro

l in

pu

t (N

ew

ton

)

Thruster 6


Figure 8.9: Application of the NASF controller (Punctual external disturbance rejection):

(a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution of the controlinputs.

−1.5

−1

−0.5

0

0 50 100 150−0.5

−0.4

−0.3

−0.2

−0.1

0

Time (s)

W

-B(N

)Gz W

(N

.m)

Figure 8.10: Application of the NASF controller (Punctual external disturbance rejec-

tion): Time history of the evolution of the parameters gr= [W−B, zGW]T .

8.5. APPLICATION OF THE L1 ADAPTIVE CONTROLLER 109

8.5 Application of the L1 adaptive controller


Tuning the parameters of the L1 adaptive controller is not a difficult task since few pa-

rameters need to be chosen. Given that the robustness and the adaptation are decoupled,

there is no need to have an a priori knowledge of the system in order to initialize the pa-

rameter vector. For this reason the vector σ and the vector θ are set to 0. Only a knowledge

of the dynamics of the system is needed in order to choose the poles of the state matrix.

For the low pass filter we have chosenD(s) = 1s which is the filter suitable for a wide class

of systems. Concerning the adaptation gains, large gains for a faster convergence are se-

lected.

Table 8.2: Parameters’ values of the L1 adaptive controller used in the experiments.

Parameter Description Value Parameter Description Valueσzinitial Initial value for the nonlinear term of z 0 kz Feedback gain of z 0.141

σϑinitial Initial value for the nonlinear term of ϑ 0 kϑ Feedback gain of ϑ 0.15

θzinitial Initial value for the parameter θ along z 0 Γz Adaptation gain of z 700000

θϑinitial Initial value for the parameter θ along ϑ 0 Γϑ Adaptation gain of ϑ 70000

D(s) Low pass filter 1s


The nonlinear adaptive state feedback controller presented in the previous section re-

vealed to have a satisfactory closed-loop behavior in the different proposed scenarios.

However, the obtained performance was achieved thanks to the approximate initialization

of the parameter vector and the adequate tuning of the PID gains. Despite the observed

robustness towards the parameter change, a small degradation in the closed-loop behav-

ior was noted. In this section, the L1 adaptive controller is tested. It was selected in order

to overcome the classical drawbacks encountered in adaptive controllers. In fact, it has

a particular architecture where robustness and adaptation are decoupled. This results in

a fast adaptation with high gains without the need of an a priori knowledge of the model

parameters.


Figure 8.11(a) displays the evolution of the vehicle’s position in depth and pitch for the

L1 adaptive controller. The initial desired depth of 40 cm is reached within 30 s. A delay


of 10 s is observed at the beginning and it is due to the presence of the filter in the control

loop. For the rest of the trajectory in depth, we observe a clear time lag with respect to

the reference trajectory. However no time lag is observed for the trajectory tracking of the

pitch angle controller but a steady state error of 1 deg can be observed at ϑ= 0 deg. The

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.11: Application of the L1 adaptive controller (Control in nominal case): (a) evo-lution of the system outputs’ responses (z and ϑ) and (b) evolution of the control inputs.

−20

0

20

40

60

80

100

Pa

ram

ete

rs(θ

)

θz

θϑ

0 50 100 150−80

−30

20

70

120

170

Time (s)

Dis

turb

an

ces

(σ)

σz

σϑ

Figure 8.12: Application of the L1 adaptive controller (Control in nominal case): Timehistory of the evolution of the parameters θ = [θz, θϑ]

T and the nonlinear terms σ =

[σz, σϑ]T .


thrusters exhibit a smooth response (cf. Figure 8.11(b)) which is interesting to mention

especially that its controlled parameters were all initialized to zero as shown in Figure 8.12

and the adaptation gains were very large. We therefore deduce that a satisfactory closed-

loop behavior is achieved without the need of any a priori knowledge of the system.


The same additional buoyancy presented for the previous controllers was added to the

system. The tracking performance of the pitch did not change in comparison with the

nominal conditions but we observed a small degradation with the depth in terms of steady

static error (cf. Figure 8.13(a)). The convergence to the first level of depth was in fact fast

(20 seconds to reach 37 cm) due to the fast convergence of the parameters (cf. 8.14) but

a steady state error of 3 cm was present and then it was replaced by a negative error of

1.5 cm. For the other levels of depth there was a positive steady state error of 1 cm. The

control inputs generated by the thrusters are depicted in Figure 8.13(b). Compared to the

nominal case, we can observe that the robot’s thrusters are exerting more effort in order to

immerse the vehicle while keeping the desired pitch angle; we have a combined force of

1.4 N compared to 1.2 N in the nominal case. However, no saturation was observed and

the same smoothness in the profiles of the control inputs was conserved.

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.13: Application of the L1 adaptive controller (Robustness towards parameter

uncertainty): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution ofthe control inputs.


−20

0

20

40

60

80

100

Pa

ram

ete

rs(θ

)

θz

θϑ

0 50 100 150−80

−30

20

70

120

170

Time (s)

Dis

turb

an

ces

(σ)

σz

σϑ

Figure 8.14: Application of the L1 adaptive controller (Robustness towards parameter

uncertainty): Time history of the evolution of the parameters θ= [θz, θϑ]T and the non-

linear terms σ= [σz, σϑ]T .


As specified earlier, an external punctual disturbance (mechanical impact) has been

applied on the vehicle and the measured positions are displayed in Figure 8.15(a). The

pitch angle for this controller was more severely affected by this disturbance but it was

able to stabilize in 5 s. The maximal overshoot was of 15 deg. Concerning the depth, dis-

turbances of 15 cm amplitude such as the one performed at t=18 s, was recovered within

4 s similarly to the one performed at time t=60 s. The largest overshoot was noted when

a disturbance occurred at time t=114 s during the variation of the reference trajectory and

it was equivalent to 12 cm. The settling time in this last case was 10 seconds instead of

5. We can also observe a small static error of 1.5 cm that is maintained after almost every

impact. Despite these above mentionned remarks, the L1 adaptive controller has a good

ability to reject the external disturbances. This can be clearly seen on the short settling

times and overshoots despite various amplitudes of the disturbances. These differences in

the system’s response of each controller are also reflected in Figure 8.15(b)where the con-

trol input of the L1 adaptive controller is seen to react smoothly with a more significant

change in its estimated parameters as seen in Figure 8.16.


0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−20

−15

−10

−5

0

5

10

15

20

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.15: Application of the L1 adaptive Controller (Punctual external disturbance

rejection): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution ofthe control inputs.

0

50

100

150

Pa

ram

ete

rs(θ

)

θz

θϑ

0 50 100 150

−50

0

50

100

150

200

Time (s)

Dis

turb

an

ces

(σ)

ˆσz

σϑ

Figure 8.16: Application of the L1 adaptive Controller (Punctual external disturbance

rejection): Time history of the evolution of the parameters θ= [θz, θϑ]T and the nonlinear

terms σ= [σz, σϑ]T .


8.6 Application of the extended L1 adaptive controller


In order to show what the PID augmentation on the originalL1 adaptive controller can

bring, the parameters used in the previous section were kept the same. However, PID gains

were chosen for each of the controlled degree of freedom (i.e depth and pitch) according

to the following:

For the depth position the gains are:

Kpz = 1500, Kiz = 200, Kdz = 500

And for the pitch angle the following were retained:

Kpϑ = 5, Kiϑ = 0.1, Kdϑ = 3.5


The extended L1 adaptive controller was designed in order to reduce the time lags ob-

served on the original L1 controller when the reference trajectory is varying. The tuning

of the PID gains only depends on how much the robustness of the system needs to be pre-

served. The higher the PID gains are, the lower the robustness becomes. Given that the

PID augmentation does not get through the filter and its aim is only to reduce the time lag,

its tuning is not a difficult task. Selecting the gains is a compromise between robustness

and a good tracking.


Figure 8.17(a) displays the evolution of the vehicle’s depth and pitch under nominal

conditions. The time lag observed with the standard L1 adaptive controller has been dras-

tically reduced and we observe that after the convergence to 0.4 m, the augmented L1

controller is almost able to perfectly track the desired trajectory. However, we note that

the static error of 1deg with the pitch angle is still preserved. The thruster consumption

is however larger compared to the original L1 controller. In fact, the noise in the control

input has a maximum amplitude of 0.2 N. No saturation occurs. The reason behind this

profile change is the presence of the PID. A compromise on the choice of these gains can

be made in order to either chose a better tracking performance or a less power consump-

tion. The larger the PID gains are, the better the tracking is on behalf of a more noisy input

profile. Comparing the performance of this controller with last three presented ones, we

8.6. APPLICATION OF THE EXTENDED L1 ADAPTIVE CONTROLLER 115

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.17: Application of the extended L1 adaptive Controller (Control in nominal

case): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution of thecontrol inputs.

−20

0

20

40

60

80

100

Pa

ram

ete

rs(θ

)

θz

θϑ

0 50 100 150−80

−30

20

70

120

170

Time (s)

Dis

turb

an

ces

(σ)

σz

σϑ

Figure 8.18: Application of the extended L1 adaptive Controller (Control in nominal

case): Time history of the evolution of the parameters θ = [θz, θϑ]T and the nonlinear

terms σ= [σz, σϑ]T .

can note that the positions have a tracking performance similar to the NASF one. How-

ever, with the augmented L1 adaptive controller, the parameters have been initialized to

zero as displayed in Figure 8.18. The absence of the necessity of an a priori knowledge


of the model with this extended version shows that the robustness of the controller was

preserved although the adaptation gains were large.


The same robustness test was performed for the extended L1 adaptive controller with

the addition of an extra buoyancy. The depth and pitch are displayed in Figure 8.19(a) and

their profile is seen to be the same as the one of the nominal conditions (Figure 8.17(a)).

This shows the robustness of this extended version to parameter change. The presence

of this added buoyancy can be however noticed in the variation of the parameters that

converged to different values without needing an additional delay or compromising the

closed-loop behavior (cf. Figure 8.20). Concerning the control inputs depicted in 8.19(b),

we can observe that the robot’s thrusters are exerting more effort in presence of the added

buoyancy. It is also interesting to mention that the oscillations in the control input do not

increase and that no thruster saturation is observed. We deduce from this scenario that this

scheme is able to ensure the same performance for the closed-loop system in presence

of a variation in the model parameters while guaranteeing a good tracking performance

despite the zero initialization of the parameter vector and the presence of large adaptation

gains.

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−10

−5

0

5

10

15

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.19: Application of the extendedL1 adaptive Controller (Robustness towards pa-

rameter uncertainty): (a) evolution of the system outputs’ responses (z and ϑ) and (b)evolution of the control inputs.

8.6. APPLICATION OF THE EXTENDED L1 ADAPTIVE CONTROLLER 117

−20

0

20

40

60

80

100

Pa

ram

ete

rs(θ

)

θz

θϑ

0 50 100 150−80

−30

20

70

120

170

Time (s)

Dis

turb

an

ces

(σ)

σz

σϑ

Figure 8.20: Application of the extendedL1 adaptive Controller (Robustness towards pa-

rameter uncertainty): Time history of the evolution of the parameters θ= [θz, θϑ]T and

the nonlinear terms σ= [σz, σϑ]T .

0

0.2

0.4

0.6

0.8

De

pth

(m

)

Desired Trajectory

Measured Depth

0 50 100 150−40

−30

−20

−10

0

10

20

30

40

Time (s)

Pit

ch a

ng

le ϑ

(d

eg

)

Desired Trajectory

Measured Pitch


0

0.5

1

1.5

Thruster 5

0 50 100 1500

0.5

1

1.5

Time (s)

Co

ntr

ol i

np

ut

(Ne

wto

n)

Thruster 6


Figure 8.21: Application of the extended L1 adaptive Controller (Punctual external dis-

turbance rejection): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evo-lution of the control inputs.


Experimental results were performed to validate the ability of the extended L1 controller

to reject disturbances. Figure 8.15(a) displays the evolution of the vehicle’s depth and pitch


−200

−100

0

100

200

Pa

ram

ete

rs(θ

)

θz

θϑ

0 50 100 150

−200

−100

0

100

200

Time (s)

Dis

turb

an

ces

(σ)

σz

σϑ

Figure 8.22: Application of the extended L1 adaptive Controller (Punctual external dis-

turbance rejection): Time history of the evolution of the parameters θ= [θz, θϑ]T and the

nonlinear terms σ= [σz, σϑ]T .

for a varying trajectory in presence of numerous disturbances occurring randomly along

the trajectory. Small disturbance amplitudes (≈ 5 cm) were injected at times t = 58 s

and t = 78 s and were recovered in 3 s and 5 s respectively with almost no overshoots.

With a disturbance of a higher amplitude the robot overshoots to the surface as seen at

times t= 108 s and t= 165 s. The reason behind these overshoots is the saturation of the

thrusters caused by the PID augmented block (cf. Figure 8.21(b)). These saturations also

cause higher overshoots (≈ 25 deg) for the pitch angle with higher settling times (≈ 10 s)

compared to 5 s recovery times and (≈ 5 deg) of overshoots for small disturbances. We

can deduce from this scenario that the thrusters tend to reach saturation when an external

disturbance is applied on the vehicle. However, it is possible to go around this problem

by reducing the PID gains. In fact, the saturation do not only occur on the control inputs

but also on the controlled parameters at ±200 as depicted in Figure 8.22. It can be seen

for instance at times t= 20 s and t= 160 s and for σz starting t= 60 s. According to the

adaptation law explained in chapter 5, it was mentioned that the parameters are bounded

using the projection law to avoid the problem of divergence and instability. In our case, the

bound was set to 200 and we can note that it was reached.

8.7 Comparison among the various proposed controllers

In the previous sections, each controller has been tested for the same scenarios. The

closed-loop system under these various schemes exhibits a different behavior that can be

8.7. COMPARISON AMONG THE VARIOUS PROPOSED CONTROLLERS 119

seen through the tracking performance, robustness and thruster consumption. To sum-

marize the experimental results previously discussed, all the figures have been displayed

per scenario for a better visualization of their differences (cf. Figure 8.23 to Figure 8.25). A

qualitative comparative study among these controllers will be therefore provided per sce-

nario. Table 8.3 presents in a quantitative manner the differences underlying these various

schemes. Some relevant criteria have been chosen to perform this comparison namely:

– settling time: the settling time is taken for each degree for freedom for the first step of

variation. According to the trajectory’s features, the first depth level is programmed

to be reached within 15 seconds, and the pitch one within 5 seconds. This criterion

helps in evaluating the convergence speed of each control scheme.

– tracking performance: after reaching the first level of variation, each controlled de-

gree of freedom is expected to follow two different steps. The tracking performance

can be seen through the presence of time lags, overshoots or steady state errors. It

was evaluated in the table in terms of the ’+’ sign. The worst tracking performance

is denoted with one ’+’ sign and the best with four ’+’ signs. This criterion has only

been evaluated once, given that it stays the same for the remaining scenarios.

– steady state maximum error: the presented control schemes do not present any no-

ticeable overshoot. For this reason, the maximum error is an indication of the max-

imum steady state error after the vehicle joined the reference trajectory. Given that

the depth levels are small, the values of the errors are represented instead of the per-

centage.

– steady state root mean square error (RMSE): it is considered the effective value of the

error. It is obtained through the root of the averaged squares of the position errors

over the whole trajectory, starting after the settling time.

– root mean square force (RMSF): similarly to the previous criteria, this one applies

to the thruster consumption. The forces exerted by both thrusters are squared,

summed and averaged. The root of this latter value gives an estimate of the aver-

age consumption provided.

– residual oscillations on the input: this criterion concerns the smoothness profile of

the control input. The values of the maximum oscillation amplitude are displayed in

the table.

– recovery time: this criterion only applies for the scenario of the external disturbance

rejection. It gives an indication of the ability of the controller to recover from the ap-

plied disturbance and follow its prescribed position. Two values have been selected.

The first one pertains to the recovery of a small to medium impact, and the second

one concerns the recovery time of a stronger impact.


– maximum overshoot: this criterion concerns the external disturbance and repre-

sents the maximum depth overshoot observed after an impact.


In nominal conditions, the gains of all the proposed schemes have been tuned and

then have kept unchanged for the rest of the experiments. As mentionned previously, three

variations in depth and in pitch are proposed. For the depth position, the best tracking ob-

served is the one with the extended L1 adaptive controller. This can be seen through the

fastest settling time (23 s) and small errors (3.8 cmmaximal error and 1.1 cmRMSE). How-

ever, we could notice that the closed-loop system controlled by the NASF exhibits a similar

behavior given that the settling time is only 2 s longer and the errors are very close (4.2 cm

of maximal error and 1.22 cm or RMSE). It can also be considered superior for the first

20 s of the response due to the negligible time lag observed. On one hand, the parameter

vector of the NASF has been initialized to some estimated values allowing the thrusters to

exert an initial force ensuring the instantaneous dive. On the other hand, the two schemes

based on the L1 controller have their parameter vector initialized to zero causing an ini-

tial delay. The initial delay with the PID controller is caused by the time needed by the

integral term to compensate for the floatability. However, its maximal error and its RMSE

are the smallest compared to all other schemes. These errors have been calculated once

the vehicle joins the desired trajectory. For this reason, when considering the trajectory

starting from the settling time, the PID controller was able to achieve the best tracking per-

formance. Nevertheless, the tracking in depth is considered satisfactory for all schemes

except for the L1 controller that exhibits a clear time lag at every depth variation. The

maximum error being 9.2 cm has been recorded at t= 53 s during the change of the tra-

jectory from 0.4m to 0.6m. The time lag is caused by the filter denoted byD(s) on Figure

5.6, that delays the convergence to the desired reference trajectory when this latter is not

constant. Concerning the pitch angle, the best tracking observed is the one related to the

PID scheme. The RMSE is 0.4 deg and the maximum error is 2 deg. However, the residual

oscillations pertained to this controller are the highest. A time lag is observed when the

NASF controller is used and a static error is observed when the pitch stabilizes at −5 deg

with the L1 based controllers. As for the control inputs, the profile of the L1 controller is

seen to be the smoothest, followed by the NASF, the extended L1 controller and finally the

PID scheme whose control input has large noise oscillations having a maximum of 0.4 N

compared to 0.1 N for the L1 adaptive controller). The root mean square force is the in-

dication of the average consumption that should be close to the floatability of the vehicle.

Indeed, we observe that all the schemes have this value close to 1.25 N. We can deduce

from this scenario, that the figures of the NASF and the extended L1 controller are very

8.7. COMPARISON AMONG THE VARIOUS PROPOSED CONTROLLERS 121

similar. However, the main advantage that the extended L1 controller holds is the absence

of the need for an a priori knowledge of the model parameters.

Scenario 2: Robustness to Modeling Uncertainties: Change in buoyancy

In this scenario, the robustness of the controllers is evaluated by adding a buoyancy

(piece of floating polyester foam) on top of the vehicle. The initial time lag in the depth po-

sition with the PID controller is seen to be more important compared to the nominal case.

However in terms of maximal errors and RMSE for the rest of the trajectory, this scheme

still holds the smallest values (2.3 cm and 1.1 cm). When comparing the performance

with the nominal conditions, we observe a less important time lag concerning the initial

variation in depth with the NASF that is almost absent when the L1 based schemes are im-

plemented. The settling time of the extendedL1 controller was still the same (23 s) and the

values of the errors remained almost the same. These criteria prove that the performance

of the closed-loop system in presence of a new parameter uncertainty is not affected un-

der an extended L1 controller. The same remark can be made for the original L1 adaptive

controller concerning the values of the errors that remained almost the same. However, an

additional delay was observed when reaching the first level of depth. This delay occurred

at time t= 25 s which means that the slope of descent of the vehicle remained the same.

As for the pitch angle, no significant change can be observed for the proposed schemes but

an important chance is noticed in the profiles of the control input. The thrusters, with the

PID controller, saturate when the vehicle is required to go down while keeping a positive

pitch. In fact, the root mean square force of all the schemes are expected to have increased

by about 0.2 N with the addition of the polyester foam. However, the value of the aver-

age force for the PID is seen to be higher than expected (1.61 N instead of 1.45 N). This

is caused by the fact that thruster 6 saturated for around half of experimental test increas-

ing unnecessarily the power consumption. We can deduce from this scenario that all the

controllers were able to overcome this imposed uncertainty but that the PID, despite its

good tracking performance, exhibited a more important initial delay due to the saturation

of the motors. The other adaptive controllers were seen to be robust to the modeling un-

certainty. The interesting remark concerning the L1 adaptive schemes is the similarity of

the closed-loop behavior observed between the nominal case and the robustness test.

Scenario 3: Rejection of External Disturbances: Mechanical Impact

In this scenario, several external disturbances of various amplitudes have been applied

to the system. The scheme seen to be able to best reject the disturbances is theL1 adaptive

controller. The recovery times are the shortest despite the fact that the amplitude of the in-


jected disturbances for this controller is the highest. The shortest recovery time was of 4 s

for a soft impact, compared to 10 s for a bigger one. In this latter case, the maximal over-

shoot was of 12 cm. The good response of the system under this controller can also be seen

through the control input where thrusters recover fast without any significant overshoot.

Analysing the history of the parameters, one can consider that the high gain based adap-

tivity of this controller probably contribute to the fast rejection. The NASF performance

in presence of disturbances is seen to be satisfactory compared to the other remaining

two schemes where the vehicle is brought to the surface when the disturbance exceeds a

certain limit. This observed maximum overshoot also coincides with the saturation of the

thrusters. In summary, theL1 adaptive controller is seen to respond the best in presence of

external disturbances followed by the NASF and last by the PID and the extended L1 con-

troller. It is important to mention that the extended L1 controller inherits characteristics

from the PID given that the augmented block is a PID scheme. An improvement could be

brought on the performance of this extended version of the L1 controller by reducing the

Kp andKi gains of the PID. It will be a compromise between the tracking performance and

the ability of rejecting disturbances.

8.8. CONCLUSION 123

8.8 Conclusion

In this chapter, real-time experiments have been performed on the underwater vehi-

cle described in chapter 7. The four control schemes explained in chapters 4, 5, and 6

have been tuned for the tracking of the depth and the pitch and tested in three differ-

ent situations to evaluate their performance under the nominal regime, their robustness

and their ability to reject disturbances. For this reason, the control parameters have been

tuned to suit the nominal case and have been kept unchanged for the rest of the exper-

iments. The performed experiments show that the NASF controller has a similar perfor-

mance compared to the extended L1 controller in terms of robustness to parameter un-

certainty and tracking precision. The NASF has a higher ability to reject disturbance com-

pared to the extended L1 controller but remains inferior for this criterion compared to the

original L1 controller. However, the tuning of the NASF is more delicate since the param-

eter vector needs an adequate initialization and the robustness towards the uncertainties

on the parameters is limited. The higher the uncertainty is, the slower the closed-loop

response will be, given that it depends on the adaptation gain that cannot be set high in

order not to harm the robustness of the system. The decoupling characteristic between

robustness and adaptation present with the L1 adaptive controller ensures the same per-

formance of the closed-loop system under various uncertainties as seen through the ex-

periments thanks to the high adaptation gains. Finally a well tuned PID leads to high

precision and very good tracking performances. However, tuning a PID is a hard work

for very small underwater vehicles like the AC-ROV. Moreover, the PID’s power consump-

tion is increased compared to the other controllers and saturations of the actuators occur

more easily. These saturations induce undesirable behaviors (large overshoots, time-lags

...). The reader can refer to the following internet link for a video with experimental results:

www.lirmm.fr/∼creuze/ocean/.


Figure 8.23: Time history of the controlled positions (depth and pitch), and the controlinputs in nominal conditions for the four proposed control schemes.

8.8. CONCLUSION 125

Figure 8.24: Time history of the controlled positions (depth and pitch), and the controlinputs in presence of a parameter change for the four proposed control schemes.


Figure 8.25: Time history of the controlled positions (depth and pitch), and the controlinputs in presence of external disturbances for the four proposed control schemes.

8.8.C

ON

CLU

SION

127

Table

8.3:Co

ntro

llers’Perform

ance

Co

mp

arison

Nominal Conditions PID NASF L1 Controller Extended L1 Controller

zzz ϑϑϑ zzz ϑϑϑ zzz ϑϑϑ zzz ϑϑϑ

Settling time 30s 5 s 25 s 7 s 28s 5 s 23 s 5 s

Steady state maximum error 3.36 cm 2 deg 4.2 cm 3 deg 9.2 cm 2.7 deg 3.8 cm 3.8 deg

Steady state root mean square error 0.9 cm 0.4 deg 1.22 cm 0.81 deg 2.7 cm 1 deg 1.1 cm 1 deg

Tracking performance ++ ++++ +++ + + ++ ++++ +++

Root mean square force1.28 N 1.26 N 1.24N 1.22 N

Maximum noise amplitude on the input0.4 N 0.1 N 0.05N 0.2 N

Robustness to parameter uncertainties

zzz ϑϑϑ zzz ϑϑϑ zzz ϑϑϑ zzz ϑϑϑ

Settling time 35 s 5 s 35 s 7 s 35 s 5 s 23 s 5 s

Steady state maximum error 2.3 cm 2.8 deg 6 cm 2.7 deg 9.2 cm 2.8deg 3.9 cm 3.2 deg

Steady state root mean square error 1.1 cm 0.57 deg 1.54 cm 0.62 deg 2.8 cm 1.1 deg 1.2 cm 1.1 deg

Root mean square force1.61 N 1.45 N 1.51N 1.5 N

External disturbance rejectionzzz ϑϑϑ zzz ϑϑϑ zzz ϑϑϑ zzz ϑϑϑ

Recovery time 10 s/40 s 5 s/10 s 8 s/10 s 4 s 4 s/10 s 4 s 5 s/15 s 5 s/10 s

Maximum overshoot surfaces 40 deg 15 cm 15 deg 12 cm 15 deg surfaces 35 deg

General Conclusion and Perspectives

The scope of this dissertation has been the design of control schemes aiming on assist-

ing the pilot of small remotely operated vehicles (also called mini ROV) through the design

of control schemes. Due to their high power over weight ratio, these vehicles are very sen-

sitive to parameter variations and external disturbances. After an overview over what was

currently available in underwater vehicle control, a special attention has been carried for

adaptive control schemes. Four different controllers have been studied and validated on

an experimental testbed for simultaneous depth and pitch control. This thesis ends here

with a summary of the work and a glimpse on the future work.

Summary of the work

Having defined the dynamic model of the vehicle and extracted the reduced model

concerning the controlled positions, four different control schemes have been proposed.

The PID controller has been initially chosen to serve as a basis for comparison since it is

considered as the most used scheme onboard underwater vehicles. Three adaptive con-

trollers have been later proposed. The first one is the well proven nonlinear adaptive state

feedback (NASF) proposed by [Fossen, 2002] and already implemented on underwater ve-

hicles. The second one is the novel L1 adaptive controller that had not been used for our

studied field yet. The particularity that this new method holds is the fact that adaptation

and robustness are decoupled. This yields a fast convergence without the necessity of hav-

ing an a priori knowledge of the system. However, its main drawback appears when the

reference trajectory is varying. The presence of a filter in its architecture induces a time

lag with respect to the desired position. This time lag has been the main motivation be-

129

130 GENERAL CONCLUSION AND PERSPECTIVES

hind the design of the extended L1 controller that we have introduced in this thesis. This

extended version holds an augmented PID block and the input generated from this block

is summed to the original control input yielding a drastic reduction in the previously ob-

served time lag. A successful stability analysis has been derived for this newly proposed

method proving that the margins of stability have been conserved for a specific choice of

PID gains. However, a degradation in the performance can be noted when these gains are

increased.

Experimental results have been performed on an experimental testbed developped in

the Laboratory of Informatics, Robotics and Microelectronic of Montpellier (LIRMM) de-

rived from the available commercial mini ROV: AC-ROV. Three different scenarios (nomi-

nal case, parameter uncertainty and external disturbance rejection) were applied for each

of the four proposed control schemes namely: PID, nonlinear adaptive state feedback

(NASF), L1 adaptive controller and extended L1 adaptive controller. The gains of each

controller have been set during the nominal conditions and were kept unchanged for the

rest of the scenarios. The performance of the closed-loop system under the PID revealed

to be satisfactory in terms of trajectory following and precision. However, the profiles of

the control input were very oscillatory and saturation was reached during the robustness

test. This induced a degradation in the closed-loop performance. In what concerns the

remaining schemes, results have shown that the NASF shows a similar behavior compared

to the extended L1 controller in terms of trajectory following and robustness to parame-

ter uncertainty. It can also be superior when it comes to the ability of rejecting external

disturbances, but its main inconvenient is the need to adequately initialize its parameter

vector. In addition to that, with the NASF, adaptation and robustness are coupled, for a

faster adaptation in presence of a large parameter variation the closed-loop performance

can be slowed down. Moreover, increasing the adaptation gain in order to increase the

convergence can destabilize the system. What is interesting to mention with the two pre-

sented L1 adaptive schemes is that their parameter vectors can be initialized to zero. Later

on, large variations in the model parameters can occur without affecting the convergence

speed of the system. This is possible thanks to the fact that adaptation and robustness

are decoupled allowing the presence of very large adaptation gains while guaranteeing the

smoothest response for the control inputs. Finally, we

Future work

This dissertation can be a first stepping stone to further elaborations on the topic. Some

perspectives can be done on the short term and others on the long term. For the short term,

we have noticed that the proposed extended L1 controller shows a weak ability to reject

FUTURE WORK 131

disturbances and could therefore be improved by replacing the augmented PID block by a

saturated PID or even another more performant scheme. The study could also be extended

to additional degrees of freedom using another prototype with a different sampling time.

For the long term, experiments in a real uncontrolled environment can be considered. The

underwater vehicle could be brought to perform a specific task such as dam inspection

using the control schemes proposed in this thesis. Vision feedback control can also be

used since such robots have a onboard camera. This tool can help in controlling more

degrees of freedom and adding more precision to the required tasks.

APPENDIX

ARoll stabilization with an internal

rotating disk

A.1 Introduction

The size of an underwater vehicle has a great impact on its control and stability. In the

case of big underwater vehicles (weighing more than 100 kg), the inertia combined to a

poor power/mass ratio contributes to increase the vehicle’s stability. Inversely, the iner-

tia of small underwater vehicles (weighing less than 15 kg) renders them more sensitive to

external disturbances (shock, hydrodynamic effects, etc.). Moreover, such a class of vehi-

cles often offers an increased power/mass ratio, thus increasing manoeuvrability but also

leading to internal disturbances due to the dynamical effects of the thrusters themselves.

The inertial counter torques as well as the gyroscopic effects produced by the motors and

the propellers induce variations in the robot’s attitude. These variations are caused by the

disturbing effects coming from the acceleration of the motors. In addition to these ef-

fects, the propeller torque can be seen to have the most important impact on the vehi-

cle’s orientation. Indeed, most of the effects listed above have been already incorporated

in various applications of aerial vehicles [Mulhaupt et al., 1999], but never been consid-

ered in underwater vehicles. The main reason behind this omission is the presence of

nonlinear hydrodynamic and viscous effects considered predominant and the usual use

of big sized vehicles attenuating the disturbances caused by the thrusters. Nevertheless,

some control methods based on gyroscopic stability and internal actuation have been

used in underwater systems. They were firstly investigated in [Leonard, 1996][Leonard,

1997a][Leonard, 1997b][Leonard et Marsden, 1997] where the behavior of the open-loop

was analyzed, and physical motivation was used to exploit geometry in order to stabilize

unstable motions. From this study, the idea to use internal actuation to stabilize an under-

water vehicle was born. A reduced model of an ellipsoidal vehicle having one or two rotors

133

134 APPENDIX A. ROLL STABILIZATION WITH AN INTERNAL ROTATING DISK

inside was initially proposed [Leonad et Woolsey, 1998]. Then, a full model was presented

and validated through simulation results [Woolsey et Leonard, 1999b][Woolsey et Leonard,

1999a][Woolsey et Leonard, 2002]. This ongoing research of internal momentum exchange

led to the development of the underwater vehicle IAUMBUS [Shlutz et Woolsey, 2003]. A

scheme for attitude control based on gyroscopic torques was presented in [Thornton et al.,

2005] and [Thornton et al., 2006]. Four control moment gyros units arranged in a pyramid

configuration were introduced inside IKURA, a zero-G vehicle (its center of buoyancy and

gravity are coincident). It was the first robot able to dive with a vertical pitch and then sur-

face in surge. Our interest is directed towards less conventional control methods of small

underactuated vehicles based on a more complete model for its dynamics. The control

method of interest falls among the less conventional schemes that would use the nonlin-

earities of the model emerging from the inertial counter torques as a mean of stabilization

rather than neglecting them. This study uses a small underwater vehicle unactuated in roll.

The proposed solution aims at stabilizing the roll while compensating for the undesirable

effects caused by the thrusters and acting on all the angles of orientation. The acceleration

of the added rotor will stabilize the roll while the disturbances acting on the pitch and yaw

are compensated in our control law via a feedforward added to a nonlinear state feedback

control law [Fossen, 2002]. We propose here a study that includes all the undesirable ef-

fects of the thrusters’ dynamics and incorporates them in the control scheme with the aim

of compensating them.

A.2 System Description

The vehicle under study is the AC-ROV already described in chapter 7. One of its pro-

pellers has been replaced in this study by a disk which acceleration is supposed to pro-

vide a torque. This torque will act not only on the pitch but also on the roll. Figure A.1(a)

shows the direction of the thrust exerted by the propellers as well as the axis of rotation of

an added inertial disk. Two transversal cuts of the robot are shown to present the system

components and their configuration. Figure A.1(b) displays the propellers and the disk,

whereas Figure A.1(c) displays the orientation of the motors. Gear trains connect each mo-

tor to its propeller under an angle of π2 . All the motors are positioned in the (x,y) plane.

In this study, we are interested in the control of the orientation only, the translation will

not be treated. The yaw control is provided thanks to the differential speed control of the

thrusters 1 and 2. Pitch control is ensured using thrusters 3 and 4, whereas the roll is not

actuated by thrusters. The rotational velocityωdisk seen in Figure A.1(a) refers to the an-

gular velocity of the motor’s rotor which makes a lead disk turn.

τ is the vector of torques produced by the thrusters to control the orientation angles with

A.3. DYNAMIC MODELING OF THE UNDERWATER VEHICLE 135

(a) (b)

disk/

p2p2p2/

p4p4p4/

p3p3p3/

p1p1p1/

(c)

mdiskmdiskmdisk/

m2m2m2

m4m4m4/

m3m3m3

m1m1m1/

Figure A.1: View of the vehicle under study with the orientation of the thrusts and the axisof the disk (a) as well as a transversal cut showing the positions of the propellers, motors,and added disk (b)(c). The body-fixed frame of reference (xbybzb) is also shown along withthe angle pertaining to each axis (b)(c).

τ= [0 τpitch τyaw]T . Therefore the control input expressed inN.m is given by:

τ= TKω|ω| (A.1)

where T ∈ R3×4 is the thrusters’ configuration matrix taking into account the position and

orientation of the propellers, allowing thus to determine the associated forces in the body-

fixed frame. K is the control input coefficient of proportionality between the angular ve-

locity and the obtained torques. ω ∈ R4 is the vector of angular velocities of the motors

actuating the four propellers, in rad.s−1.

A.3 Dynamic Modeling of the Underwater Vehicle

A.3.1 Background

The dynamic model used follows the SNAME notation and the representation described

in section 3.3 and reminded below. It is expressed for the degrees of freedom under study:

roll, pitch, and yaw.

η= J(Θ)(η)ν (A.2)

Mν+C(ν)ν+D(ν)ν+g(η)= τ+wd (A.3)

whereν= [p,q,r]T ,η= [φ,θ,ψ]T are vectors of angular velocities (in the body-fixed frame)

and Euler angles (in the earth-fixed frame) respectively. J(Θ) ∈ R3×3 is the transforma-

tion matrix mapping the body-fixed angular velocities to the earth-fixed ones. The model

matrices M(η), C(η), and D(η) ∈ R3 denote inertia (including added mass), Coriolis-

centripetal (including added mass), and damping respectively, while g(η) is a vector of


gravitational/buoyancy forces. τ ∈ R3 is the vector of control inputs acting only on pitch

and yaw (τroll = 0 N). wd ∈ R3 is the vector of disturbances to be detailed here after. In

the case of our study, the vehicle used has a slow dynamics, and hence it will be moving at

velocities low enough to make the Coriolis terms negligible (C(ν)≈ 0).

A.3.2 Disturbance effects

In the dynamical model (A.3), external disturbances coming from the environment are not

taken into account and therefore wd only holds the undesirable effects induced by the

thrusters’ dynamics. The impact of these effects is put upfront in this study by considering

their models and incorporating them in the vehicle’s model. Then, the vectorwd takes the

following form:

wd=−τctm−τctp−Q−τGyrom−τGyrop (A.4)

The first two terms represent the inertial counter torques of the motors and propellers re-

spectively. Q is the propeller load torque and the last two terms are the gyroscopic effects

produced by the motors and the propellers. Given the configuration of the motors and

propellers, we notice that their axes of rotation do not coincide with the ones of the robot.

For this reason, all the torques calculated are projected into the robot’s frame. This adds

coupling and complexity to the system. The details of their computation is given here be-

low:

– Inertial counter torques: This term appears on the rotational axis of each motor and

propeller given that they have different axes as illustrated in Figure A.1(b) and Figure

A.1(c). It occurs upon a change in the rotational velocity of the motor and propeller

generating an opposing resisting torque on the vehicle. It is given by:

τctmi = Jmωi for each motor i (i= 1...4) and for the disk’s motor, with Jm the rotor

inertia and ωi the time derivative of its angular velocity. Combining the effects of all

the motors and projecting them along the robot’s axes, we get:

τctm = [τctmroll τctmpitch τctmyaw] with

τctmroll = cos(π4 )(τctm1−τctm2+τctm3−τctm4+τctmdisk)

τctmpitch = sin(π4 )(−τctm1+τctm2+τctmdisk)

τctmyaw = 0

τctmyaw is null since the motors are in the (x,y) plane and therefore they have

no effect on the yaw. Similarly, the same equation is applied for each propeller:

τctpi = Jpωi

Gratiowith i (i = 1...4), Jp being the propeller inertia and Gratio the gear

ratio between the motor and the propeller. The expression of the vector τctp is not

A.4. PROPOSED CONTROL SCHEME 137

explicitly written since it can be obtained by performing similar projections as for the

case of the motors.

– Propeller load torque: The propeller load torque acts on the system in the opposite

sense of the propeller angular velocity. It is given as a function of the thrusters’

parameters that will be incorporated in our study in a lumped parameter Kq. We

therefore get: Qi = Kq|ωi

Gratio|ωi

Gratio, and Q = [cos(π4 )(Q1−Q2), sin(π4 )(Q1 −

Q2), −Q3−Q4]T .

– Gyroscopic torques: These disturbances are caused by the gyroscopic effect induced

when a change in the angular momentum of the motors or propellers occurs. It is

given by: τGyromi = Jmωi∧ν for the motors with i (i= 1...4) and for the disk’s motor,

ν= [p,q,r]T being the rotational velocity of the vehicle. Similarly for the propellers

we have: τGyropi = Jpωi

Gratio∧ν with i (i= 1...4) . The evaluation of this effect have

demonstrated that it is negligible compared to other disturbances and therefore, it

will be neglected.

A.4 Proposed Control Scheme

The model presented in (A.3) is subjected to various disturbances emanating from its ac-

ωm

ωdisk

ω= [ωmωdisk]T

ωm= [ω1ω2ω3ω4]T

τdes

ω

η,η, η

ηdes, ηdes, ηdes

ν

τdesf

Figure A.2: Block diagram of the proposed control scheme.

tuation and movements. These undesirable effects have an impact on the performances

of the closed-loop system especially when dealing with a small vehicle. Our objective is to

design and implement a control law that will take into account these disturbances and im-


prove the behavior of the vehicle in closed-loop. The proposed control scheme illustrated

through the block diagram of Figure A.2 is described in three main parts, namely:

1. Nonlinear State Feedback Control: This controller, as suggested in [Fossen, 2002], is

applied to the actuated variables of the orientation vector (θ andψ).

2. Roll stabilization: In the absence of external disturbances, oscillating effects are ex-

pected to appear on roll given the coupling between the degrees of freedom due to

the configuration of the motors shown in Figure A.1. An internal rotor with a disk

is then incorporated to compensate the effects generated by the others motors with

the use of its inertial counter torque.

3. Feedforward Control: This part incorporates the calculated compensation along

pitch and yaw of the undesirable torques produced by the thrusters and the com-

pensation of the disturbing effects along the pitch thanks to the rotation of the above

mentioned disk.

In the following, these three parts of the control scheme will be detailed.

A.4.1 Nonlinear State Feedback Control

The proposed nonlinear state feedback controller is that suggested in [Fossen, 2002] which

is based on the linearization of the commanded acceleration an for a trajectory following

in the earth frame. To guarantee that the error converges to zero, an is then chosen as the

following Proportional Integral Derivative (PID) control:

an= ηdes−KPη−KI

∫ t

0

ηdt−KD ˙η (A.5)

with η = η−ηdes and ˙η is its first derivative, ηdes is the desired trajectory and ηdes is its

second derivative. The computed input is calculated in the body-fixed frame but the tra-

jectory following is performed in the earth-fixed frame and therefore ab, the acceleration

in the body-fixed frame, is calculated from the simple following transformation:

ab= T−1r (η)(an− Tr(η) ν) (A.6)

Introducing equation (A.6) in the dynamic model (A.3), the control law that cancels the

nonlinearities is then chosen to be:

τdes=Mab+C(ν)ν+D(ν)ν+g(η) (A.7)

τdes being the desired torques and forces to be applied on the vehicle. One remarks that

wd, the disturbance term explained in (A.3) is not yet taken into account but will be com-

pensated through the feedforwad explained below yielding the final vector τdesf .

A.5. NUMERICAL SIMULATIONS 139

A.4.2 Roll Compensation

We propose to use the acceleration of the disk’s motor as a control input to induce a torque

that cancels the effects on the roll. The torque τroll provided by the disk should be equal

in magnitude to the disturbing effects and also opposite in direction in order to ensure a

complete compensation. From the definitions and equations provided in section 3.B, we

get:

τroll=−(τctm3 −τctm4 )−cos(π4 )(τctm1 −τctm2 −τctp1 +τctp2 −Q1+Q2)

Taking into account the configuration of the motor and disk, the inertial counter torque

produced by the disk along the roll axis is then expressed by:

cos(π4 )(−JdiskGratio

ωdisk+Jmωdisk) = τroll

We extract from this last equation the acceleration ωdisk to be applied in order to compen-

sate the effects induced on the roll and stabilize it. However, in many underwater vehicles,

(this is the case of the AC-ROV for instance) size constraints do not allow the disk’s axis to

be parallel to the x axis of the vehicle. This implies a coupled effect on roll and pitch. This

problem is overcome thanks to the proposed feedforward described in the sequel.

A.4.3 Feedforward for Pitch and Yaw

The vector τdes ∈ R3 is described in (A.7). The vector of angular velocitiesω can therefore

be deduced from (A.1) and it will be used for the computation of the feedforward control to

be summed with τdes in order to compensate the disturbing effects induced by the motors.

Similarly to the compensation of the roll performed above, we compute the torques needed

on the pitch and yaw:

– Feedforward for the pitch control input:

τffθ =−sin(π4 )(−τctm1 +τctm2 −τctp1 +τctp2 +τctdisk−Q1+Q2)

– Feedforward for the yaw control input:

τffψ =−τctp3 −τctp4 +Q3+Q4

All the terms have been detailed in the previous section. It is worth to note that τctdisk is

the counter torque induced by the disk’s rotor. We will finally get:

τff= [0 τffθ τffψ]T and the final expression of the control input becomes τdesf = τdes+τff.

A.5 Numerical Simulations

The efficiency of the proposed control scheme is put at stake in the following numerical

simulations by displaying the behavior of the closed-loop system’s orientations with and

without the compensation of the disturbance effects described in section 3. The input

model parameters used in the numerical simulations are summarized in table A.1. Simula-


tions have been performed in Matlab software with the well-proven mss simulator [Fossen

et Perez, 1991]. The obtained results are displayed in Figure A.3 along with the control in-

puts being the torques of the motors (cf. Figure A.4). A desired oscillating trajectory in yaw

is generated to put upfront the disturbances on the roll and pitch. The latter two degrees of

freedom are stabilized around 0 deg. Three scenarios have then been performed, namely:

1. Scenario 1: Nonlinear State Feedback applied on yaw and pitch: An oscillating de-

sired trajectory is imposed on the yaw while the pitch is controlled to remain stable

at 0 deg. No action is taken on the roll and the effects of the thrusters’ dynamics are

observed on the pitch and roll.

2. Scenario 2: Nonlinear State Feedback applied on yaw and pitch with roll stabiliza-

tion: Similarly to the previous case, the same trajectory is applied on the yaw and

pitch. However, the roll is stabilized using the added disk by compensating the com-

puted disturbances on this degree of freedom.

3. Scenario 3: Proposed Controller: In this scenario we apply the controller detailed

in section 4 and observe the behavior of the closed-loop system when performing

the same trajectory as in scenario 2. The pitch and the yaw are controlled to follow a

desired trajectory using the proposed nonlinear state feedback controller augmented

by a feedforward compensating the disturbances caused by thrusters’ dynamics. The

roll is stabilized by the added disk similarly to the previous scenario.

4. Scenario 4: Gyroscopic effects and disk size: In this scenario, the proposed controller

is again applied with the same desired trajectories on the studied degrees of freedom.

The gyroscopic effects neglected until now will be taken into account and the size of

the disk will progressively be increased. The aim of this scenario is to observe how

these new considerations can affect the behavior of the closed-loop system.

Table A.1: Input model parameters values used in simulations.

Parameter Description Value

Jm Rotor inertia 5.7×10−7kg.m2

Jp Propeller inertia 1.59×10−6kg.m2

Kq Propeller torque coefficient 9.25×10−8N.m.rad−1.s

Jdisksmall Small disk inertia 3.52×10−5kg.m2

Jdiskmedium Medium disk inertia 4.56×10−5kg.m2

Jdiskbig Big disk inertia 5.1×10−5kg.m2


A.5.1 Scenario 1: Nonlinear State Feedback applied on the yaw and

pitch

The aim of this scenario is to show the induced effects of the motors and propellers on

the roll and pitch when the roll is not controlled. The pitch is controlled to remain stable

around 0 deg while a desired trajectory oscillating from −45 deg to +45 deg is imposed

on the yaw. This persistent oscillation is intentionally made to observe the induced dis-

turbances on the remaning degrees of freedom in orientation. Figure A.3(a) displays the

evolution of the three orientation angles under study. We observe that the yaw follows

the desired trajectory in closed-loop under the nonlinear state feedback controller, while

the pitch despite being controlled, exhibits some minor residual oscillations ranging from

−0.6 deg to +0.6 deg. The roll is left uncontrolled and therefore oscillates from −4 to

+4.5 deg. Figure A.4(a) shows the torques of the motors. Motors 1 and 2 controlling the

yaw provide torques of around +1 N.m and −1 N.m, motors 3 and 4 stabilizing the pitch

exhibit torques between +0.1N.m and −0.05N.m. In this scenario the disk’s motor is kept

off. Figure A.5(a) shows the rotational velocities of the motors. Motors 1 and 2 controlling

the yaw have an angular velocity of around 1100 rad/s whereas motors 3 and 4 stabilizing

the pitch have angular velocities of around 300 rad/s.

−6

−4

−2

0

2

4

6

Roll

(Deg)

−6

−4

−2

0

2

4

6

Pit

ch(D

eg)

0 10 20 30 40 50−60

−40

−20

0

20

40

60

Time (s)

Yaw

(Deg)

(a) Scenario 1: Nonlinearstate feedback applied onyaw and pitch

−6

−4

−2

0

2

4

6

Roll

(Deg)

−6

−4

−2

0

2

4

6

Pit

ch(D

eg)

0 10 20 30 40 50−60

−40

−20

0

20

40

60

Time (s)

Yaw

(Deg)

(b) Scenario 2: Nonlinearstate feedback applied onyaw and pitch with roll sta-bilization

−6

−4

−2

0

2

4

6

Roll

(Deg)

−6

−4

−2

0

2

4

6

Pit

ch(D

eg)

0 10 20 30 40 50−60

−40

−20

0

20

40

60

Time (s)

Yaw

(Deg)

(c) Scenario 3: Proposedcontroller

Figure A.3: Time history of the measured orientation angles in blue as well as the desiredtrajectories in yaw and pitch in dotted red lines for the three scenarios.


A.5.2 Scenario 2: Nonlinear State Feedback applied on the yaw and

pitch with disk-based roll stabilization

In this scenario, the objective is to control all the degrees of freedom pertaining to the ori-

entation. The desired trajectory in yaw is the same as the previous scenario while the pitch

is stabilized around 0 deg. These two degrees of freedom are controlled using the nonlin-

ear state feedback controller. The roll is stabilized thanks to the effect of the incorporated

disk’s acceleration through a feedforward that cancels out the disturbances caused by the

coupled effects of the thursters’ dynamics. Figure A.3(b) displays the evolution of the ori-

entation angles. We can see clearly that the amplitude of the oscillations is increased in

the closed loop response of the pitch angle (from −1.2 deg to +1.3 deg). This is due to

the effect of the disk and its corresponding rotor that are not taken into account and that

significantly disturb the pitch. However, one observes that the effects on the roll were re-

duced by half compared to the previous scenario, that is, current oscillations are ranging

from −1.7 deg to +2.5 deg. The torques of the motors 1 and 2 in this scenario are similar

to the previous case (cf. Figure A.4(b)), motors 3 and 4 controlling the pitch increase their

consumption to torques varying from −0.15 N.m to 0.15 N.m. The motor’s disk has the

most important consumption since its counter torque varies from −1.3 Nm to +1.3 N.m.

The rotational velocities of the motors 1 and 2 in this scenario are similar to the previous

case (cf. Figure A.5(b)), motors 3 and 4 controlling the pitch have an increased rotational

velocity of 500 rad/s. The motor’s disk has the most important consumption since it satu-

rates at the maximum allowed rotational velocity being 1500 rad/s.

A.5.3 Scenario 3: Proposed Control Scheme

The results of the proposed control scheme, detailed in section 4, are displayed in Figure

A. 3(c). The difference with respect to the previous scenario is that a feedforward has been

added on the pitch and yaw. The roll angle exhibits a similar behavior compared to the

previous case and the yaw angle still follows the desired trajectory. However, the pitch an-

gle is stabilized close to 0 deg. The thrusters’ effects are therefore completely compensated

on this latter degree of freedom with the addition of the feedforward. The motors have a

very similar consumption compared to the previous scenario (cf. Figure A.4(c)). The con-

sumption of motors 3 and 4 increases slightly and now vary from −0.2 N.m to 0.2 N.m. It

results then that the motors also have a very similar rotational velocity compared to the

previous scenario (cf. Figure A.5(c)). The rotational velocities of motors 3 and 4 increased

to 600 rad/s.


−1

0

1

τ1

(N.m

)

−1

0

1

τ2

(N.m

)

−1

0

1

τ3

(N.m

)

−1

0

1

τ4

(N.m

)

0 10 20 30 40 50

−1

0

1

Time (s)

τdisk

(N.m

)

(a)

−1

0

1

τ1

(N.m

)

−1

0

1

τ2

(N.m

)−1

0

1

τ3

(N.m

)

−1

0

1

τ4

(N.m

)

0 10 20 30 40 50

−1

0

1

Time (s)

τdisk

(N.m

)

(b)

−1

0

1

τ1

(N.m

)

−1

0

1

τ2

(N.m

)

−1

0

1

τ3

(N.m

)

−1

0

1

τ4

(N.m

)

0 10 20 30 40 50

−1

0

1

Time (s)

τdisk

(N.m

)

(c)Figure A.4: Torques of the thrusters in (a) the first scenario, (b) the second scenario and (c)the third scenario.

(a) (b) (c)Figure A.5: Angular velocities of the motors in (a) the first scenario, (b) the second scenarioand (c) the third scenario.

A.5.4 Scenario 4: Gyroscopic effects and disk size

It has to be noticed that with a bigger disk a better stabilization of roll can be expected.

Figure A.6 shows the evolution of the roll angle for three different disk sizes. It was stated

in section 3 that the gyroscopic effects are negligible which was applicable in the absence

of the internal disk or in the presence of a small one. The bigger the disk is, the more iner-

tia he has, and since it is turning at a large angular velocity, its gyroscopic effect becomes

important because the disk’s inertia increases. Figure A.7 displays the roll angle when the

proposed control law was applied in the case of a small disk (cf. Figure A.7(a) ) and a big

disk (cf. Figure A.7(b)). The red dotted lines in Figure A.7 refer to the case with gyroscopic

effects in the model and the blue solid lines refer to the case when these effects are ne-

glected. We observe that the peak to peak amplitude of the roll angle is the same when we

have a small disk (peak to peak amplitude of 4 degrees) even though the oscillations are


0 2 4 6 8 10 12 14 16 18 20

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time(s)

φ (d

eg)

Small DiskMedium DiskBig Disk

Figure A.6: Time history of the roll angle depending on the disk’s inertia

0 5 10 15 20−5

−4

−3

−2

−1

0

1

2

3

4

5

Time(s)

φ (d

eg)

Without Gyro EffectsWith Gyro Effects

(a)

0 5 10 15 20−5

−4

−3

−2

−1

0

1

2

3

4

5

Time(s)

φ (d

eg)

Without Gyro EffectsWith Gyro Effects

(b)Figure A.7: Roll angle in presence of a small disk (a) and a big disk (b) with overlappingplots both neglecting or not the gyroscopic effects.

shifted. However, in presence of a big disk, we notice that when the gyroscopic effects are

taken into account the roll angle’s peak to peak amplitude increased five times (the oscil-

lations varying from −0.3 deg to +0.3 deg increased to −1.8 deg to +1.2 deg). This obser-

vation is important and should be considered when designing the disk. Indeed, increasing

the disk’s inertia will not completely cancel out the roll oscillation as the gyroscopic effects

are no more negligible.

A.6. CONCLUSION 145

A.6 Conclusion

This chapter introduces a novel concept for roll stabilization of an underactuated under-

water vehicle. A new control architecture for the orientation of an underactuated under-

water vehicle is presented. It is based on a nonlinear state feedback controller augmented

by a feedforward control for the pitch and yaw. The unactuated roll is stabilized using the

inertial counter torques induced by an internal motor on which a disk was added. A com-

plete study of the disturbances emanating from the motors and affecting the robot’s orien-

tation was presented. Numerical simulations have shown the effectiveness of the proposed

scheme through the obtained promising results.

APPENDIX

BProof of stability of the NASF

For the nonlinear adaptive state feedback controller, the error dynamics is guaranteed

to converge to zero by applying Barbalat’s lemma on a chosen Lyapunov function [Fossen,

2002]. The design parameters needed to satisfy the stability are the positive constants c0and c1 defined in 4.11 in the equation of the combined error along with the proportional,

derivative and integral positive constants (Kp,KdandKi) present in the linearization of the

commanded accceleration. The PID gains can be chosen by using pole placement whereas

the constants c0 and c1 should satisfy the requirements below:

(i) (c0Kd+c1Kp)c1> c20I

(ii) 2c0Kp >βI

(iii) 2(c1Kd−c0I)>βI

β is taken as a small positive constant. The PID gains can be taken as diagonal matrices

according to the dimension needed and I is the identity matrix of the dimension involved.

147

APPENDIX

CProof of stability of the AC-ROV with

the L1 adaptive controller

For the proof of stability we consider (5.29) under the following general formalism:[η1

η2

]=Am

[η1

η2

]+

[02×21M∗r

]ωua+g(t,X(t)) (C.1)

with X = [η1,η2], g(t,X(t)) = [02, g2(t,X(t)]T where g2(t,X(t)) is the nonlinear func-

tion being the gravitational and buoyancy forces and uncertainty on damping and mass:

g2(t,X(t))=

(W−B)M∗z

+w∗dzM∗z

−WzGcos(ϕ)sin(ϑ)

M∗ϑ

+w∗dϑM∗ϑ

+

[−∆

D∗z

M∗zz

−∆D∗ϑ

M∗ϑϑ

].

The symbol ∆ refers to the uncertainty on these parameters affecting each studied degree

of freedom and being present on the diagonal matrix ofA2. All the other terms have been

explained previously.

Some assumptions and terms need to be defined in order to prove the stability of the

system:

Remark: g2(t,X(t)) and M∗r also depend on ϕ the roll angle for the transformation

between the body frame and the earth frame. The vehicle is stable in this degree of free-

dom and not actuated and for this reason it has been neglected.

Assumption 1: Boundedness ofg2(t,0)g2(t,0)g2(t,0): There existsB> 0 such that ||g2(t,0)||∞ ≤B.

g2(t,0)=

(W−B)M∗z

+w∗dzM∗z

w∗dϑM∗ϑ

.

M∗z being around 3Kg , the largest element in this matrix is

w∗dϑM∗ϑ

sinceM∗ϑ is of the order

of 0.05Kg.m2 and w∗dϑ

could have a maximal value of 1N.m which is a very conservative

149

150APPENDIX C. PROOF OF STABILITY OF THE AC-ROV WITH THE L1 ADAPTIVE

CONTROLLER

value of a disturbance acting on the pitch. We then considerB= 25, being the conservative

bound on ||g2(t,0)||∞.

Assumption 2: Semiglobal uniform boundedness of partial derivatives of g2(t,X)g2(t,X)g2(t,X): For

an arbitrary δ > 0, there exist positive constants dg2x (δ) > 0 and dg2t (δ)> 0 independent

of time such that for all ||X(t)||∞ < δ, the partial derivatives of g2(t,X(t)) are piecewise

continuous and bounded:

g2(t,X(t)) is in fact independent of time, so we are left with the partial derivative with

respect to the state as following:∣∣∣∣∣∣∂g2(t,X(t))∂X

∣∣∣∣∣∣∞

=

∣∣∣∣∣

∣∣∣∣∣

[0 0

−Wrgzcos(φ)cos(ϑ)

Mϑ(ϑ)+

w∗dϑ

Mϑ(ϑ)0

]∣∣∣∣∣

∣∣∣∣∣∞

Mϑ(ϑ) is the term designating the partial derivative ofM∗ϑ with respective to ϑ. This new

term is also of the same order as M∗ϑ. Therefore with the partial derivatives of g2(t,X(t))

given as such, we can conclude that it is uniformly bounded with dg2x =B= 25.

Let us define the following variables needed for the condition of stability:

XinXinXin is defined such as Xin(s) = s(I−Am)−1X0, with X0 being the values of X at t= 0

ρ0ρ0ρ0 is defined such as ||xin|| ≤ ρ0. The robot being initialized at the surface from a static

position, a conservative choice of ρ0= 0.1 is chosen. It refers to a maximum of 0.1rad and

10cm.

ρinρinρin= ||s(sI−Am)−1||L1ρ0= 0.2042.

ρrρrρr= 3.2042 (defined according to the following: ρr > ρin and therefore it was chosen to be

ρr= ρin+3).

ρρρ= ρr+ γ1 with γ1 being an arbitrary small constant. We choose ρ= ρr+0.1= 3.3042.

LρrLρrLρr =ρ.dgXρr

= 25.7802.

rrr is the reference trajectory to be tracked.

KgKgKg is a feedforward prefilter with Kg=−(CA−1m Bm)

−1.

The following two transfer functions are also needed:

G(s) =H(s)(I2−C(s)) and H(s) = (sI2−Am)−1Bm (C.2)

With a choice of the transfer functionD(s)= 1sI2 we get the low pass filter to be:

C(s) =ωK(sI2+ωK)−1 (C.3)

151

Finally, as stated in [Hovakimyan et Cao, 2010] the sufficient condition of stability

needed to be fulfilled is the one below:

||G(s)||L1 <ρr− ||kgC(s)H(s)||L1 ||r||L∞

−ρin

Lρrρr+B(C.4)

Replacing with the above given numerical values, we get:

0.0175< 0.0186

The matlab code below holds all the calculation details of the L1 norms of the transfer

functions along with the numerical values of the model parameters. The definition of the

L1 norm can be found in the appendix related to the mathematical tools.


CONTROLLER

%Proof of stability of the underwater vehicle in depth and pitch

clc

clear all

%Model parameters Ac-Rov

D11=14;

D22=13;

D33=12;

D44=0.15;

D55=0.17/4; %increase error on D and compensate it with theta, we are not

%sure of the parameter anyways, and having the imposed one renders our

%matrix easily transformed into hurwitz (like this we might also have a

%better estimate)

D66=0.18;

Ix= 3*(0.149^2+0.152^2)/12;

Iy=3*(0.154^2+0.152^2)/12;

Iz=3*(0.149^2+0.154^2)/12;

M11=3;

M22=3;

M33=3;

M44=Ix;

M55=Iy*4;

M66=Iz;

A=[ 0 0 1 0

0 0 0 1

0 0 -D33/M33 0

0 0 0 -D55/M55]

Bm=[ 0 0

0 0

1/M33 0

0 1/M55]*0.01

153

km=[150 0 20 0; 0 10 0 15]

Am=A-Bm*km

eig(Am)

w=100/156.6; %multiply w by 100 and divide B by 100, it is the same thing,

%it is a way of normalizing to keep w between 0 and 1 , a good thing is

%that our parameter will be reduced by a factor of 100 since Bm multiplies

%theta and sigma

s=tf(’s’)

I=eye(4);

c=[1 0 0 0

0 1 0 0 ]

k=30*eye(2);

Ds=eye(2)*1/s

r_linf=1

%calculation of rhoin by computing L1 norm

trfunc=s*inv(s*I-Am);

rho_0=0.1

B_0=25 %very conservative bound, this is the bound of f(t,0) for all t so

%it is an external disturbance..of max 5 N when all x is 0

for i=1:4 %rows of transfer matrix

for j=1:4 %columns

[ss,rr]=impulse(trfunc(i,j),0:0.01:15);

rhoint(i,j)=sum(abs(ss)) ;

rhoint(i,j)=rhoint(i,j)*rr(2)

end

end

L1norm_rs=zeros(4,1);

for ii=1:4 %rows of transfer matrix

for jj=1:4 %columns


CONTROLLER

L1norm_rs(ii)=rhoint(ii,jj)+ L1norm_rs(ii);

end

end

rhoin= max(L1norm_rs)*rho_0

rho_r= rhoin+3;

dfx=25;

gama_bar=0.1 %random constant

L_rhor=(rho_r+gama_bar)*dfx/rho_r

%%L1 norm of kgCsHs

kg=-inv(c*inv(Am)*Bm)

Hxm=(s*I-Am)^(-1)*Bm

Cs=w*k*(s*eye(2)+w*k)^(-1);

HxCsKg= Hxm*Cs*kg

%G(s)

Gs=Hxm*(eye(2)-Cs)

%Norm G(s)L_rhor


for j=1:2 %columns

[aa,bb]=impulse(Gs(i,j),0:0.01:15);

155

Norm_Gs_p(i,j)=sum(abs(aa)) ;

Norm_Gs_p(i,j)= Norm_Gs_p(i,j)*bb(2);

end

end

L1norm_rs=zeros(4,1);


for jj=1:2 %columns

L1norm_rs(ii)= Norm_Gs_p(ii,jj)+ L1norm_rs(ii);

end

end

Norm_Gm=max(L1norm_rs)

%Norm HxCsKg


for j=1:2 %columns

[cc,dd]=impulse(HxCsKg(i,j),0:0.01:15);

Norm_Hxck_p(i,j)=sum(abs(cc)) ;

Norm_Hxck_p(i,j)= Norm_Hxck_p(i,j)*dd(2);

end

end

L1norm_Hxck=zeros(4,1);


for jj=1:2 %columns

L1norm_Hxck(ii)= Norm_Hxck_p(ii,jj)+ L1norm_Hxck(ii);

end

end


CONTROLLER

Condition=(rho_r - max(L1norm_Hxck)*r_linf - rhoin)/(L_rhor*rho_r+B_0)

APPENDIX

DUseful Mathematical Tools

In this chapter the important mathematical tools used for the establishment of the the

control schemes are briefly reminded.

D.1 Infinity Norm

D.1.1 Vector

The ∞ norm of a vector is the maximum element in the vector in absolute value. Its

mathematical expression for a vector x of n elements is the following:

||x||∞ = max (|x1|, |x2|, ..., |xn|) (D.1)

D.1.2 Matrix

The ∞ norm of a matrix X ∈Rn×m is defined as the maximum element of the vector

created by the summation of the absolute values of the elements of the rows. It can be

expressed according to the following:

||X||∞ = max1≤i≤n

m∑

j=1

|xij| (D.2)

D.2 L1 Norm

For the study of the stability, we are interested in the norms of functions. For this rea-

son the focus will be towards the L1 norm of the transfer functions mapping an input to

157

158 APPENDIX D. USEFUL MATHEMATICAL TOOLS

an output. Without loss of generality, we will therefore consider bounded input bounded

output linear time varying systems. We consider a system G(s) in Laplace form with m

inputs and l outputs. The impulse response of this system is denoted by g(t,t0).

The L1 norm is therefore given by:

||G||L1 = max1≤i≤m

l∑

j=1

||gij||L1 (D.3)

where

||gij||L1 = supt≥τ,τ∈R+

∫ t

τ

|gij|(t,σ)dσ (D.4)

gij is the (i, j) entry of the impulse response matrix.

D.3 Projection Operator

For adaptive schemes, it is becoming more common to encounter the projection oper-

ator in order to prevent parameter drift. It is based on theory from convex analysis. For the

sake of simplicity, we will only provide the definition of the projection operator. The reader

if interested could look into convex analysis for further details.

We consider a smooth convex function defined by:

f(θ)=(ǫθ+1)θ

Tθ−θ2maxǫθθ2max

(D.5)

θmax is the bound imposed on the vectorθ (maximum value any element ofθ can take).

ǫθ > 0 is the projection tolerance chosen. The projection operator is then defined as:

Proj(θ,y) =

y if f(θ)< 0

y if f(θ)≥ 0 and ∇fTy≤ 0

y− ∇f||∇f||

⟨ ∇f||∇f||

,y⟩f(θ) if f(θ)≥ 0 and ∇fTy> 0

(D.6)

with ⟨∇f||∇f||

,y⟩ denoting the dot product between ∇f||∇f||

and y.

APPENDIX

EDetails of the model’s parameters

In this chapter, we give the numerical values of the model matrices used for the robot

when performing the simulations and the experiments.

We therefore remind the reader with the dynamical model of the underwater vehicle

AC-ROV :

Mν+C(ν)ν+D(ν)ν+g(η)= τ+wd (E.1)

The model matrices needed to be determined are the mass matrixM, the dampingD

and the gravitational and buoyancy forces g. The Coriolis termC is neglected.

Since the identification of the model parameters is not in the scope of this thesis, these

parameters have been initialized with a very rough approximation in order to be close to

the dynamics of the system. The uncertainties present will also highlight more the neces-

sity as well as the robustness and performance of the designed adaptive schemes.

Here below are the expressions and the numerical values of the matricesM, D and g.

The assumptions and the a priori information taken into account are the following:

(i) The vehicle has a cubic shape and weighs 3 Kg.

(ii) The dimensions of the robot are the following: heighth= 15.2 cm, length l= 15.4 cm

and widthwi= 14.9 cm.

(iii) The vehicle moves at low velocities, the added mass and the Coriolis effects will be

neglected.

(iv) The vehicle is symmetric and therefore the model matricesM andD will be consid-

ered symmetric.

159

160 APPENDIX E. DETAILS OF THE MODEL’S PARAMETERS

We consider the underwater vehicle in its 6 degrees of freedom:

The mass matrix is given by:

M=

m 0 0 0 0 0

0 m 0 0 0 0

0 0 m 0 0 0

0 0 0 Ix 0 0

0 0 0 0 Iy 0

0 0 0 0 0 Iz

m denotes the mass of the vehicle, Ix, Iy and Iz denotes the moments of inertia around the

x,y and z axis respectively. Given the rectangular shape of the vehicle these moments of

inertia are given by the following, as we assume that the mass of the ROV is homogeneously

distributed.

Ix=m.(wi×h)/12

Iy=m.(l×h)/12

Iz=m.(wi× l)/12

Replacing with the numerical values, we get:

M=

3 0 0 0 0 0

0 3 0 0 0 0

0 0 3 0 0 0

0 0 0 0.0113 0 0

0 0 0 0 0.0117 0

0 0 0 0 0 0.0115

The damping matrixD is symmetric and approximated using a rough estimate of the

maximal speed of the vehicle in the water, it is given by:

D=

14 0 0 0 0 0

0 13 0 0 0 0

0 0 12 0 0 0

0 0 0 0.15 0 0

0 0 0 0 0.17 0

0 0 0 0 0 0.18

Finally the matrix g representing the gravitational and buoyancy forces is given by:

161

g=

−sin(ϑ).(B−W)

cos(ϑ).sin(ϕ).(B−W)

cos(ϕ).cos(ϑ).(B−W)

W.zG.cos(ϑ).sin(ϕ)

W.zG.sin(ϑ)

0

As seen beforeB denotes the buoyancy andW the weight. The Euler angles for roll and

pitch are respectivelyϕ and ϑ. The origin of the fixed-body reference frame is the center of

buoyancy. For the determination of the buoyancy and gravitational forces, we are left with

the z coordinate of the gravity center rgz . Given then the AC-ROV is neutrally buoyant we

have B=W = 30 N. With rgz = 0.02mwe get:

g=

0

0

0

0.6.cos(ϑ).sin(ϕ)

0.6.sin(ϑ)

0

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Robots, 24:7–24, 2000. Cited pages 6 and 16.

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Junky Yuh et Jing Nie : Application of non-regressor-based adaptive control to underwater

robots: experiment. Computers and Electrical Engineering, 26:169–179, February 2000.

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List of Figures

1.1 Example of mini ROVs used for inspection.(Courtesy of AC-CESS, Seabotix and

Ocean Modules) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Comparative table among some commercial mini ROVs . . . . . . . . . . . . . . . 7

1.3 An example of a trajectory for automated dam inspection by an underwater ve-

hicle. Systemic scanning using constant intervals of depth.[Maalouf et al., 2012b] 8

1.4 Total floating production storage and off loading (http://www.sjcho.com/) . . . 9

1.5 Two approaches of hip hull inspection using horizontal or vertical slices [J. Va-

ganay, 2005] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Marine drilling riser (http://oilandgastechnologies.wordpress.com/2012/08/27/steel-

catenary-risers-scr/) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7 Underwater vehicle for cleaning of nets [Borovic et al., 2011] . . . . . . . . . . . . 10

2.1 Classification of the main control schemes in underwater robotics . . . . . . . . 17

2.2 Block diagram of the PID controller proposed in [Perrier et Canudas-De-Wit,

1996] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Depth and pitch control algorithm for the AUV Autosub-1 [McPhail et Pebody,

1997] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 PD controller with fuzzy-tuned series compensation [Ostafichuk, 2004] . . . . . 20

2.5 Cascade control configuration for altitude control [Roche et al., 2011] . . . . . . 22

2.6 Framework of a µ analysis based robust scheme [Campa et al., 1998] . . . . . . . 24

2.7 Direct adaptive control method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Indirect adaptive control method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

173

174 List of Figures

3.1 View of an underwater vehicle and its reference frames (xiyizi: earth-fixed

frame, xbybzb: body-fixed frame). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Schematic view of a propeller with the representation of the vehicle speed u,

the axial flow velocity ua, the propeller disk areaAP and the generated thrust T . 42

3.3 Schematic view of the flow model proposed in [Kim et Chung, 2006] . . . . . . . 44

4.1 Block diagram of the PID based control loop. . . . . . . . . . . . . . . . . . . . . . 50

4.2 Graphical parameter estimation of an integrator model . . . . . . . . . . . . . . . 51

4.3 Graphical illustration of the step response . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Block diagram of the nonlinear adaptive state feedback controller . . . . . . . . . 54

5.1 Block diagram of the control loop based on the direct MRAC architecture . . . . 62

5.2 Block diagram of the control loop based on the direct MRAC architecture with

state predictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Block diagram of the control loop based on the L1 adaptive controller . . . . . . 62

5.4 Block diagram of the closed-loop L1 adaptive controller . . . . . . . . . . . . . . 63

5.5 Block diagram of the control loop based on the L1 adaptive controller . . . . . . 65

5.6 Block diagram of the control-loop based on the L1 adaptive control scheme . . 69

6.1 Simulated example of the tracking performance of the originalL1 adaptive con-

troller: the desired trajectory is displayed in dashed line and the output of the

controlled system in solid line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2 Block diagram of the proposed extended L1 adaptive controller . . . . . . . . . . 76

6.3 Simulated performance of the L1 adaptive controller compared with the pro-

posed extended L1 controller: the desired trajectory is displayed in red dashed

line, the L1 adaptive controller in blue solid line and the proposed extended

controller in black solid line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4 Open-loop system with the proposed extended L1 adaptive controller for a lin-

ear system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.5 Nyquist plot of the system (6.10) corresponding to the case of the original L1controller (solide blue line) and (6.9) corresponding to the PID based extended

one (dotted red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.6 Nyquist plots of the open-loop system for the case of the PID based extended

L1 controller for different values of the proportional gain: Kp = 3 in solid blue

line, Kp= 15 in dashed green line, and Kp= 30 in red dotted line. . . . . . . . . . 82


L1 for different values of the integral gain: KI = 0.5 in solid blue line, KI = 2.5

in dashed green line, and KI= 5 in red dotted line. . . . . . . . . . . . . . . . . . . 82

List of Figures 175


L1 controller for different values of the derivative gain: KD = 0.1 in solid blue

line, KD= 0.2 in dashed green line, and KD= 0.3 in red dotted line. . . . . . . . 82

6.9 Nyquist plot of the system given in (6.9) for different values of the adaptation

gain : Γ = 5000 (solid blue line), Γ = 10000, (dashed green line) and Γ = 100000

(dotted red line.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.1 View of the AC-ROV vehicle (a) and its commercial setup (b) . . . . . . . . . . . . 90

7.2 View of the vehicle under study with the orientation of the thrusts (a) as well as

a transversal cut showing the positions of the propellers and motors (b). . . . . . 91

7.3 Transversal cut of the underwater vehicle and its body-fixed frame (xbybzb).

Rotational directions of the propellers are depicted by yellow arrows. The blue

propellers have a positive blade pitch, (i.e right hand rotation) and the yellow

ones have a negative blade pitch (i.e left hand rotation). . . . . . . . . . . . . . . . 92

7.4 Thrusters’ charactersitics (a) and added buoy on top of the AC-ROV to remedy

for the hysteresis and nonlinearities (b). . . . . . . . . . . . . . . . . . . . . . . . . 94

7.5 Schematic view of the hardware architecture of the AC-ROV prototype. . . . . . 95

7.6 View of the AC-ROV experimental testbed: 1© Control PC, 2© Power input, 3©

Emergency stop button, 4© Video in, 5© Tether plug, 6© Ethernet plug, 7© Video

Capture, 8© Tether, 9© AC-ROV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.1 View of the AC-ROV in different operating conditions: (a): Nominal case, (b):

Buoyancy change, (c): External disturbance (Mechanical Impact). . . . . . . . . . 98

8.2 Application of the PID controller (Control in nominal case): (a) evolution of

the system outputs’ responses (z and ϑ) and (b) evolution of the control inputs. 100

8.3 Application of the PID controller (Robustness towards parameter uncer-

tainty): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evo-

lution of the control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.4 Application of the PID controller (Punctual external disturbance rejection):

(a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution of the

control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.5 Application of the NASF controller (Control in nominal case): (a) evolution of

the system outputs’ responses (z and ϑ) and (b) evolution of the control inputs. 105

8.6 Application of the NASF controller (Control in nominal case): Time history of

the estimated parameters gr= [W−B, zGW]T . . . . . . . . . . . . . . . . . . . . 105

8.7 Application of the NASF controller (Robustness towards parameter uncer-

tainty): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evo-


176 List of Figures

8.8 Application of the NASF controller (Robustness towards parameter uncer-

tainty): Time history of the evolution of the parameters gr= [W−B, zGW]T . . 107

8.9 Application of the NASF controller (Punctual external disturbance rejection):


control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.10 Application of the NASF controller (Punctual external disturbance rejection):

Time history of the evolution of the parameters gr= [W−B, zGW]T . . . . . . . 108

8.11 Application of theL1 adaptive controller (Control in nominal case): (a) evolu-

tion of the system outputs’ responses (z and ϑ) and (b) evolution of the control

inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.12 Application of theL1 adaptive controller (Control in nominal case): Time his-

tory of the evolution of the parameters θ= [θz, θϑ]T and the nonlinear terms

σ= [σz, σϑ]T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.13 Application of the L1 adaptive controller (Robustness towards parameter un-

certainty): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evo-


8.14 Application of the L1 adaptive controller (Robustness towards parameter un-

certainty): Time history of the evolution of the parameters θ = [θz, θϑ]T and

the nonlinear terms σ= [σz, σϑ]T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.15 Application of the L1 adaptive Controller (Punctual external disturbance re-

jection): (a) evolution of the system outputs’ responses (z and ϑ) and (b) evolu-

tion of the control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.16 Application of the L1 adaptive Controller (Punctual external disturbance re-

jection): Time history of the evolution of the parameters θ= [θz, θϑ]T and the

nonlinear terms σ= [σz, σϑ]T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.17 Application of the extendedL1 adaptive Controller (Control in nominal case):


control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.18 Application of the extendedL1 adaptive Controller (Control in nominal case):

Time history of the evolution of the parameters θ= [θz, θϑ]T and the nonlinear

terms σ= [σz, σϑ]T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.19 Application of the extended L1 adaptive Controller (Robustness towards pa-

rameter uncertainty): (a) evolution of the system outputs’ responses (z and ϑ)

and (b) evolution of the control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.20 Application of the extended L1 adaptive Controller (Robustness towards pa-

rameter uncertainty): Time history of the evolution of the parameters θ =

[θz, θϑ]T and the nonlinear terms σ= [σz, σϑ]

T . . . . . . . . . . . . . . . . . . . 117

List of Figures 177

8.21 Application of the extendedL1 adaptive Controller (Punctual external distur-

bance rejection): (a) evolution of the system outputs’ responses (z and ϑ) and

(b) evolution of the control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.22 Application of the extendedL1 adaptive Controller (Punctual external distur-

bance rejection): Time history of the evolution of the parameters θ= [θz, θϑ]T

and the nonlinear terms σ= [σz, σϑ]T . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.23 Time history of the controlled positions (depth and pitch), and the control in-

puts in nominal conditions for the four proposed control schemes. . . . . . . . . 124


puts in presence of a parameter change for the four proposed control schemes. 125


puts in presence of external disturbances for the four proposed control schemes. 126

A.1 View of the vehicle under study with the orientation of the thrusts and the axis of

the disk (a) as well as a transversal cut showing the positions of the propellers,

motors, and added disk (b)(c). The body-fixed frame of reference (xbybzb) is

also shown along with the angle pertaining to each axis (b)(c). . . . . . . . . . . . 135

A.2 Block diagram of the proposed control scheme. . . . . . . . . . . . . . . . . . . . . 137

A.3 Time history of the measured orientation angles in blue as well as the desired

trajectories in yaw and pitch in dotted red lines for the three scenarios. . . . . . . 141

A.4 Torques of the thrusters in (a) the first scenario, (b) the second scenario and (c)

the third scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

A.5 Angular velocities of the motors in (a) the first scenario, (b) the second scenario

and (c) the third scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

A.6 Time history of the roll angle depending on the disk’s inertia . . . . . . . . . . . . 144

A.7 Roll angle in presence of a small disk (a) and a big disk (b) with overlapping plots

both neglecting or not the gyroscopic effects. . . . . . . . . . . . . . . . . . . . . . 144

List of Tables

3.1 The SNAME notation for marine vessels . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Nomenclature of the notations used in this section . . . . . . . . . . . . . . . . . . 41

4.1 Coefficients of the PID controller for the integral plus dead time plants . . . . . . 52

6.1 Comparison of the stability margins for both controllers . . . . . . . . . . . . . . 80

6.2 Effects of changing the proportional gain on the stability margins. . . . . . . . . 81

6.3 Effects of changing the integral gain on the stability margins. . . . . . . . . . . . . 83

6.4 Effects of changing the derivative gain on the stability margins. . . . . . . . . . . 83

6.5 Effects of changing the adaptation gain on the stability margins. . . . . . . . . . 83

7.1 Main characteristics of the AC-ROV . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8.1 Parameters’ values of the NASF controller used in the experiments. . . . . . . . . 104

8.2 Parameters’ values of the L1 adaptive controller used in the experiments. . . . . 109

8.3 Controllers’ Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 127

A.1 Input model parameters values used in simulations. . . . . . . . . . . . . . . . . . 140

178

Abstract

Underwater vehicles have gained an increased interest in the last decades given the mul-

tiple tasks they can accomplish in various fields, ranging from scientific to industrial and

military applications. In this thesis, we are particularly interested in the category of vehi-

cles having a high power to weight ratio. Different challenges in autonomous control of

such highly unstable systems arise from the inherent nonlinearities and the time varying

behavior of their dynamics. These challenges can be increased by the low inertia of this

class of vehicles combined with their powerful actuation. A self tuning controller is there-

fore required in order to avoid any performance degradation during a specific mission.

The closed-loop system is expected to compensate for different kinds of disturbances or

changes in the model parameters. To solve this problem, we propose in this work the de-

sign, analysis and experimental validation of different control schemes on an underwater

vehicle. Classical methods are initially proposed, namely the PID controller and the non-

linear adaptive state feedback (NASF) one, followed by two more advanced schemes based

on the recently developed L1 adaptive controller. This last method stands out among the

other developed ones in its particular architecture where robustness and adaptation are

decoupled. In this thesis, the original L1 adaptive controller has been designed and suc-

cessfully validated then an extended version of it is proposed in order to deal with the ob-

served time lags occurring in presence of a varying reference trajectory. The stability of

this latter controller is then analysed and real-time experimental results for different oper-

ating conditions are presented and discussed for each proposed controller, assessing their

performance and robustness.

Keywords: Adaptive control, Underwater robotics, Nonlinear systems

180 List of Tables

Résumé

L’utilisation des véhicules sous-marins (ROV, AUV, gliders) s’est considérablement ac-

crue ces dernières décennies, aussi bien dans le domaine de l’offshore ou de l’océanogra-

phie, que pour des applications militaires. Dans cette thèse, nous abordons le problème

particulier de la commande des véhicules sous-marins à faible inertie et fort rapport puis-

sance/inertie. Ces derniers constituent des systèmes fortement non linéaires, dont la dy-

namique est susceptible de varier au cours du temps (charge embarquée, caractéristiques

des propulseurs, variation de salinité...) et qui sont très sensibles aux perturbations envi-

ronnementales (chocs, traction sur l’ombilical...). Afin d’assurer des performances de suivi

de trajectoire satisfaisantes, il est nécessaire d’avoir recours à une commande adaptative

qui compense les incertitudes ou les variations des paramètres du modèle dynamique,

mais également qui rejette les perturbations, telles que les chocs. A cette fin, nous propo-

sons dans ce manuscrit, l’étude théorique et la validation expérimentale de plusieurs lois

de commande pour véhicules sous-marins. Nous analysons tout d’abord des approches

classiques dans ce domaine (commande PID et commande par retour d’état non linéaire),

puis nous les comparons avec deux autres architectures de commande. La première est la

commande adaptative L1 non linéaire, introduite en 2010 notamment pour la commande

des véhicules aériens, et implémentée pour la première fois sur un véhicule sous-marin. Le

découplage entre adaptation et robustesse permet l’utilisation de très grands gains d’adap-

tation (et donc une convergence plus rapide des paramètres estimés, sans aucune connais-

sance a priori), sans pour autant dégrader la stabilité. La seconde méthode, que nous pro-

posons et qui constitue l’apport principal de cette thèse, est une évolution de la commande

L1, permettant d’en améliorer les performances lors du suivi d’une trajectoire variable.

Nous présentons une analyse de stabilité de cette commande, ainsi que sa comparaison

expérimentale avec les autres lois de commande (commande PID, commande adaptative

par retour d’état non linéaire et commande adaptative L1 standard). Ces expérimentations

ont été réalisées sur un mini-ROV et plusieurs scenarii ont été étudiés, permettant ainsi

d’évaluer, pour chaque loi, sa robustesse et son aptitude à rejeter les perturbations.

Mots clefs : Commande adaptative, Robotique sous-marine, Systèmes nonlineaires

LIRMM — 161, rue Ada — 34095 Montpellier cedex 5 — France

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