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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
8.4.2 Real-time experimental results . . . . . . . . . . . . . . . . . . . . . . . 103
8.5 Application of the L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . 109
vi CONTENTS
8.5.1 Controllers’ parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.5.2 Real-time experimental results . . . . . . . . . . . . . . . . . . . . . . . 109
8.6 Application of the extended L1 adaptive controller . . . . . . . . . . . . . . . 114
8.6.1 Controllers’ parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.6.2 Real-time experimental results . . . . . . . . . . . . . . . . . . . . . . . 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-
8 CHAPTER 1. CONTEXT AND PROBLEM FORMULATION
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
10 CHAPTER 1. CONTEXT AND PROBLEM FORMULATION
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-
12 CHAPTER 1. CONTEXT AND PROBLEM FORMULATION
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-
14 CHAPTER 1. CONTEXT AND PROBLEM FORMULATION
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
18 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES
– 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]
20 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES
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-
22 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES
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
24 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES
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
26 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES
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
28 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES
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-
30 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES
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
32 CHAPTER 2. STATE OF THE ART ON CONTROL SCHEMES
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)
38 CHAPTER 3. MODELING OF UNDERWATER VEHICLES
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)
40 CHAPTER 3. MODELING OF UNDERWATER VEHICLES
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
42 CHAPTER 3. MODELING OF UNDERWATER VEHICLES
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
44 CHAPTER 3. MODELING OF UNDERWATER VEHICLES
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
52 CHAPTER 4. SOLUTION 1: CONVENTIONAL CONTROLLERS
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)
54 CHAPTER 4. SOLUTION 1: CONVENTIONAL CONTROLLERS
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)
56 CHAPTER 4. SOLUTION 1: CONVENTIONAL CONTROLLERS
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)
60 CHAPTER 5. SOLUTION 2: NONLINEAR L1 ADAPTIVE CONTROLLER
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.
62 CHAPTER 5. SOLUTION 2: NONLINEAR L1 ADAPTIVE CONTROLLER
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)
64 CHAPTER 5. SOLUTION 2: NONLINEAR L1 ADAPTIVE CONTROLLER
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.
66 CHAPTER 5. SOLUTION 2: NONLINEAR L1 ADAPTIVE CONTROLLER
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
68 CHAPTER 5. SOLUTION 2: NONLINEAR L1 ADAPTIVE CONTROLLER
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
70 CHAPTER 5. SOLUTION 2: NONLINEAR L1 ADAPTIVE CONTROLLER
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
72 CHAPTER 5. SOLUTION 2: NONLINEAR L1 ADAPTIVE CONTROLLER
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:
76 CHAPTER 6. SOLUTION 3: A NEW EXTENSION OF L1 ADAPTIVE CONTROL
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
78 CHAPTER 6. SOLUTION 3: A NEW EXTENSION OF L1 ADAPTIVE CONTROL
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:
80 CHAPTER 6. SOLUTION 3: A NEW EXTENSION OF L1 ADAPTIVE CONTROL
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.
6.4. STABILITY ANALYSIS OF THE EXTENDED L1 ADAPTIVE CONTROL 81
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
82 CHAPTER 6. SOLUTION 3: A NEW EXTENSION OF L1 ADAPTIVE CONTROL
-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.
6.4. STABILITY ANALYSIS OF THE EXTENDED L1 ADAPTIVE CONTROL 83
Table 6.3: Effects of changing the integral gain on the stability margins.
Parameter Value Parameter Value Parameter Value
KP 3 KP 3 KP 3
KI 0.5 KI 2.5 KI 5
KD 0.2 KD 0.2 KD 0.2
PM 100 deg PM 92 deg PM 84 deg
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.
Parameter Value Parameter Value Parameter Value
KP 3 KP 3 KP 3
KI 0.5 KI 0.5 KI 0.5
KD 0.1 KD 0.2 KD 0.3
PM 94 deg PM 100 deg PM 106 deg
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
84 CHAPTER 6. SOLUTION 3: A NEW EXTENSION OF L1 ADAPTIVE CONTROL
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:
92 CHAPTER 7. EXPERIMENTAL CASE STUDY: THE AC-ROV UNDERWATER VEHICLE
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
94 CHAPTER 7. EXPERIMENTAL CASE STUDY: THE AC-ROV UNDERWATER VEHICLE
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
96 CHAPTER 7. EXPERIMENTAL CASE STUDY: THE AC-ROV UNDERWATER VEHICLE
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:
100 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
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
(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 8.3: 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.
102 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
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
(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 8.4: 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.
8.4.2 Real-time experimental results
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
104 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
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.
Scenario 1: Control in Nominal Conditions
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-
8.4. APPLICATION OF THE NASF CONTROLLER 105
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 8.5: 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.
Scenario 2: Robustness towards Modeling Uncertainties: Change in buoyancy
Depth and pitch obtained with the NASF controller under this scenario are depicted in
106 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
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
(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 8.7: 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.
8.4. APPLICATION OF THE NASF CONTROLLER 107
−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 .
Scenario 3: External Disturbance Rejection: Mechanical Impact
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.
108 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
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
(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)C
on
tro
l in
pu
t (N
ew
ton
)
Thruster 6
(b) Time history of the forces generated bythe two thrusters controlling z and ϑ.
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
8.5.1 Controllers’ parameters
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
8.5.2 Real-time experimental results
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.
Scenario 1: Control in Nominal Conditions
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
110 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
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
(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 8.11: 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 .
8.5. APPLICATION OF THE L1 ADAPTIVE CONTROLLER 111
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.
Scenario 2: Robustness towards Modeling Uncertainties: Change in buoyancy
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
(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 8.13: 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.
112 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
−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 .
Scenario 3: External Disturbance Rejection: Mechanical Impact
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.
8.5. APPLICATION OF THE L1 ADAPTIVE CONTROLLER 113
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
(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 8.15: 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 .
114 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
8.6 Application of the extended L1 adaptive controller
8.6.1 Controllers’ parameters
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
8.6.2 Real-time experimental results
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.
Scenario 1: Control in Nominal Conditions
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
(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 8.17: 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
116 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
of the model with this extended version shows that the robustness of the controller was
preserved although the adaptation gains were large.
Scenario 2: Robustness towards Modeling Uncertainties: Change in buoyancy
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
(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 8.19: 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
(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 8.21: 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.
Scenario 3: External Disturbance Rejection: Mechanical Impact
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
118 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
−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.
120 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
– maximum overshoot: this criterion concerns the external disturbance and repre-
sents the maximum depth overshoot observed after an impact.
Scenario 1: Control in Nominal Conditions
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-
122 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
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/.
124 CHAPTER 8. EXPERIMENTAL RESULTS OF THE PROPOSED CONTROL SCHEMES
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.
126 CHAPTER 8. EXPERIMENTAL RESULTS OF THE 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
136 APPENDIX A. ROLL STABILIZATION WITH AN INTERNAL ROTATING DISK
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-
138 APPENDIX A. ROLL STABILIZATION WITH AN INTERNAL ROTATING DISK
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-
140 APPENDIX A. ROLL STABILIZATION WITH AN INTERNAL ROTATING DISK
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. NUMERICAL SIMULATIONS 141
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.
142 APPENDIX A. ROLL STABILIZATION WITH AN INTERNAL ROTATING DISK
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.
A.5. NUMERICAL SIMULATIONS 143
−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
144 APPENDIX A. ROLL STABILIZATION WITH AN INTERNAL ROTATING DISK
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.
152APPENDIX C. PROOF OF STABILITY OF THE AC-ROV WITH THE L1 ADAPTIVE
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
154APPENDIX C. PROOF OF STABILITY OF THE AC-ROV WITH THE L1 ADAPTIVE
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 i=1:4 %rows of transfer matrix
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 ii=1:4 %rows of transfer matrix
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 i=1:4 %rows of transfer matrix
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 ii=1:4 %rows of transfer matrix
for jj=1:2 %columns
L1norm_Hxck(ii)= Norm_Hxck_p(ii,jj)+ L1norm_Hxck(ii);
end
end
156APPENDIX C. PROOF OF STABILITY OF THE AC-ROV WITH THE L1 ADAPTIVE
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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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
6.7 Nyquist plots of the open-loop system for the case of the PID based extended
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
6.8 Nyquist plots of the open-loop system for the case of the PID based extended
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-
lution of the control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
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):
(a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution of the
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-
lution of the control inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
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):
(a) evolution of the system outputs’ responses (z and ϑ) and (b) evolution of the
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
8.24 Time history of the controlled positions (depth and pitch), and the control in-
puts in presence of a parameter change for the four proposed control schemes. 125
8.25 Time history of the controlled positions (depth and pitch), and the control in-
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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