Geometric control for autonomous underwater vehicles: overcoming a
thruster failure
Michael Andonian, Dario Cazzaro, Luca Invernizzi, Monique Chyba, Sergio Grammatico
Abstract— The goal of this paper is to show how geometric
control theory can be used to design efficient trajectories for
an autonomous underwater vehicle descending into a basin,
as well as performing its recovery after experiencing an
actuator failure. The underwater vehicle is modeled as a forced
affine connection control system, and the control strategies
are developed through the use of integral curves of rank one
and kinematic reductions. As it will be presented, such a
method is particularly efficient in case of actuator failure and
it provides a constructive way to design trajectories for the new
under-actuated system. A typical scenario of basin descent is
presented, control signals are computed to realize the desired
trajectories and some simulations are provided.
INTRODUCTION
In recent years, Autonomous Underwater Vehicles (AUVs)
have allowed researchers to explore underwater environments
too hostile and dangerous for man or manned vehicles.
As technology exponentially advances, the sophistication of
AUVs grows as well. Such projects as the NASA funded
Deep Phreatic THermal eXplorer (DEPTHX) and its second
generation, the Environmentally Non-Disturbing Under-ice
Robotic ANtarctiC Explorer (ENDURANCE) are examples
of such state-of-the-art AUVs. Both projects involved de-
ploying an AUV to survey, exploiting Simultaneous Local-
ization And Mapping techniques (SLAM), the underwater
environment of Lake Bonney in Antarctica (ENDURANCE)
and a group of five sinkholes of Sistema Zacatn (DEPTHX),
in preparation and anticipation for the opportunity to explore
Europa, a moon of Jupiter, searching for life in its icy oceans
[1]. Other approaches to explore and study hostile underwater
environments include strategies involving teams of AUVs
to survey hydrothermal vents [2], by using sonar sensors
to create maps, eventually to photograph hydrothermal vent
sites [3], and by predetermining trajectories for the AUV
to follow while sampling the water to choose a viable site
to study [4]. For a long-duration mission, during which the
AUV does not have the possibility to recharge its batteries,
it is absolutely critical to take the energy demands of the
vehicle into consideration [5]. Moreover, the fact remains
that the environments AUVs have to explore are hazardous
to the vehicles as well. Precautionary techniques are therefore
M. Andonian, M. Chyba are with the Mathematics Department, College
of Natural Sciences, University of Hawai‘i, Honolulu, HI 96822, USA.
fandonian, mchybag@math.hawaii.edu
D. Cazzaro, L. Invernizzi are with the College of Engineering, Sant’Anna
School of Advanced Studies, Pisa, PI, 56127, Italy. fd.cazzaro,
l.invernizzig@sssup.it
S. Grammatico is with the Department of Electrical Systems and Au-
tomation, College of Engineering, University of Pisa, PI, 56127, Italy.
s.grammatico@dsea.unipi.it
implemented to help the vehicle to return to its starting point
safely and intact, in case of unexpected damages during
the mission. Even so, some works have placed emphasis on
fault-accommodating allocation of thruster forces on AUVs,
for instance exploiting the excess number of thrusters to
accommodate some faults during operation, see [6], [7] as
survey references. Instead the focus of this paper is the
discussion of the unexpected under-actuation during the
vehicle’s mission, from the geometric control theory point
of view. More precisely, the designed mission for the AUV
is to descent into a basin, map as accurately as possible the
walls of the cave and find its way back after an actuator
failure occured. Since the AUV has to map the walls of
an underwater cave, performing efficient trajectories that
trade off between the magnitude of the maped area and the
minimisation of the energy consumption is a priority. This is
a complex optimal control problem since the cost function
cannot be expressed in terms of the control function or the
trajectory exclusively. We here design trajectories using the
geometric framework to produce simple motions that can be
implemented on a real vehicle and that are efficicent with
respect to the goals of the mission. The layout of the paper
is as follows. The equations of motion of a submerged rigid
body are developed from a differential geometry perspective
in Section I. The resulting dynamic equations are then used
to compute the control strategies in Section II, by using
geometric methods previously established in [8], [9], with
the difference that here the affine connection control system
does not neglect potential forces. Section III provides the
simulated mission. In the last section we conclude the paper
and summarise our work.
I. AUTONOMOUS UNDERWATER VEHICLE: THE MODEL
A classic reference in marine technology for the formula-
tion of the equations of motion for an underwater vehicle
is [10]. Also [11] provides a classic framework for the
study of underwater robots. However, here we follow a
different approach to exploit the inherent geometric structure
associated to our control system. We view our AUV as a
submerged rigid body in a viscous and incompressible fluid
with rotational flow whose actuation comes in the form of
thrusters, internal motors or other sources. In the following,
only the basic calculations for deriving the complete model
are explicitly presented, while the full account of the deriva-
tion of the dynamic equations of a mechanical (underwater)
system from the geometric perspective, which goes beyond
the purposes of this work, can be found in [12] and in the
references therein.