Special Issue Article
Switching control of an underwater
glider with independently controllable
wings
Proc IMechE Part M:
J Engineering for the Maritime Environment
2014, Vol. 228(2) 136–145
! IMechE 2014
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DOI: 10.1177/1475090213502001
pim.sagepub.com
Andrea Caiti1, Vincenzo Calabrò2, Stefano Geluardi3,
Sergio Grammatico4 and Andrea Munafò5
Abstract
The dynamic, control-oriented model of an underwater glider with independently controllable wings is presented. The
onboard vehicle’s actuators are a ballast tank and two hydrodynamic wings. A control strategy is proposed to improve
the vehicle’s maneuverability. In particular, a switching control law, together with a backstepping feedback scheme, is
designed to limit the energy-inefficient actions of the ballast tank and hence to enforce efficient maneuvers. The case
study considered here is an underwater vehicle with hydrodynamic wings behind its main hull. This unusual structure is
motivated by the recently introduced concept of the underwater wave glider, which is a vehicle capable of both surface
and underwater navigation. The proposed control strategy is validated via numerical simulations, in which the simulated
vehicle has to perform three-dimensional path-following maneuvers.
Keywords
Autonomous underwater vehicle, glider, wave glider, path-following
Date received: 18 March 2013; accepted: 22 July 2013
Introduction
Underwater gliders are winged autonomous underwater vehicles (AUVs), which are usually employed for
long-term and flexible missions, unlike thrusters’ propelled underwater vehicles, such as oceanographic sampling and underwater surveillance. We indeed remind
that the underwater glider concept1 motivated the
development of several prototypes, for instance,
‘‘Slocum,’’2‘‘Spray’’3 and ‘‘Seaglider.’’4 These vehicles
are all buoyancy-propelled fixed-winged gliders and
their attitude is controlled by shifting the internal ballast. See, for instance, the control-oriented approach
developed in Leonard and Graver.5
As underwater glider can be operated in highefficiency conditions of reliance on the battery power,
an accurate control system is crucial for their performance and, especially, endurance. Most importantly,
the use of feedback control can provide robustness to
uncertainty and disturbances.5 Efficient state-feedback
control schemes are proposed in Bhatta and Leonard6
and Mahmoudian and Woolsey7 for underwater gliders
having fixed hydrodynamic wings.
More recently, the need for higher maneuverability
motivated the development of underwater gliders with
movable rudder8 or even with independently controllable wings.9 For instance, in Arima et al.,10 many experimental tests are performed with the glider ‘‘Alex’’ in
order to characterize the so-called lateral response. This
latter characteristic is clearly not available if the glider
is equipped with hydrodynamic wings that are fixed.
This study is motivated by the recently introduced
concept of the underwater wave glider,11 capable of
both wave-propelled surface navigation and buoyancydriven underwater motions. The mentioned vehicle is
indeed a hybrid underwater vehicle, in the sense that it
is both a surface vehicle and an underwater glider. Its
structure is such that the hydrodynamic wings are
1
Department of Information Engineering, University of Pisa, Pisa, Italy
Cybernetics R&D, Kongsberg Maritime, Kongsberg, Norway
3
Max Planck Institute for Biological Cybernetics, Tübingen, Germany
4
Automatic Control Laboratory, ETH Zurich, Zurich, Switzerland
5
NATO Centre for Maritime Research and Experimentation, La Spezia,
Italy
2
Corresponding author:
Andrea Caiti, Department of Information Engineering, University of Pisa,
56122, Pisa, Italy.
Email: a.caiti@dsea.unipi.it
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Caiti et al.
137
positioned in the back of its main hull when the vehicle
operates as underwater glider.
Through this article, we present a three-dimensional
(3D) path-following problem,12,13 for our vehicle, and
from a control-oriented point of view, we propose a
state-feedback switching control scheme. This article is
organized as follows. Section ‘‘Underwater glider
model’’ presents the modeling of the vehicle and the
characterization of the actuators, namely, the ballast
and the independently controlled wings. Section
‘‘Underwater gliding control problem’’ presents a
switching control scheme for a path-following maneuver. In order to validate the proposed control strategy,
many numerical simulations are performed and hence
presented in section ‘‘Simulations.’’ The last section
concludes the article.
Underwater glider model
In this section, we model the dynamics of the underwater glider, with particular emphasis on the functional
characterization of the control actuators. The vehicle is
characterized by a variable mass m(t) and a center of
gravity (CoG) with variable position rg (t) due to the
(variable) quantity of water inside the ballast tank,
positioned on top of the vehicle’s hull.
For any generic vector v, we adopt the notation with
superscript ‘‘b’’ to indicate the components of v
expressed in the body-fixed frame ‘‘attached’’ to the
vehicle, while we use the superscript ‘‘n’’ to indicate the
navigation reference frame. According to Caiti et al.,11
the equations governing the dynamics of the position
and the velocity of the CoG are
rbg (t) =
_
_
Y(t)
L + Y(t) b
m(t)
!
, r_g (t) =
rb (t)
m(t) m(t) g
m(t)
ð1Þ
where L and Y(t) are, respectively, the static and the
dynamic contributes of the CoG dynamics.
Considering the generalized position h(t) 2 R6 (with
respect to an inertial frame), the generalized velocity
v(t) 2 R6 (with respect to the body-fixed frame) and the
mass of water e(t) 2 R contained in the ballast tank, a
13-dimensional model for the rigid body motion,
including the ‘‘ballast’s dynamics,’’ can be written as
follows
M(e)v_ + C(v, e)v + D(v)v + g(h, e) + T(v, e)e_ = t
h_ = J(h)v
e_ = ue
ð2Þ
ð3Þ
ð4Þ
where the mass matrix M(e) and the Coriolis matrix
C(e) already include the added mass effects,14D(v)v is
the hydrodynamic drag force acting on vehicle, g(h, e)
indicates the forces of gravity and buoyancy and the
term T(v, e)e_ represents the effects of the variable CoG
(1). In equation (4), we assume that we can control the
velocity of the CoG e_ via the control input ue . The generalized force t consists of the (controlled)
Table 1. Numerical parameters of the considered underwater
glider.
Characteristic glider parameters
Vehicle
Length
Diameter
Static weight, ms
Static CoG position, (L=ms )x
Static CoG position, (L=ms )y
Static CoG position, (L=ms )z
Ballast tank
Capacity, max eb
Maximum rate, e_b jmax
Position, (rbb )x
Position, (rbb )y
Position, (rbb )z
Hydrodynamic wings
Width
Length
Position, (rwb )x
Position, (rwb )y
Position, (rwb )z
2m
0.15 m
23.65 kg
20.02 m
0m
0.1 m
1 kg
0.1 kg/s
0.9 m
0m
0.1 m
0.25 m
0.10 m
20.8 m
60.2 m
0m
CoG: center of gravity.
hydrodynamic effects of the wings, characterized in
detail later on. The matrix J(h) is the Jacobian matrix.
This modeling technique for vehicles having a variable position of the CoG has been also used in Caiti
and Calabrò15 for a hybrid AUV; see also Caffaz
et al.16 for further details. In Table 1, we show the
numerical parameters used in this work.
Ballast tank characterization
We assume that the position of the CoG can vary only
by filling and emptying the ballast tank. This means
that the position of the CoG, that is, rbg , can be written
as a static function of the mass of water e contained in
the ballast tank.
Based on the numerical parameters given in Table 1,
we provide a technical characterization of the ballast
actuator, positioned on top of the vehicle’s hull.
Therefore, injecting water in the ballast tank will result
in moving the CoG toward the top of the vehicle and
vice versa.
This choice also allows to regulate the pitch angle of
the vehicle by changing the water contained in the ballast tank (Figure 1). As a consequence, also the glide
angle of the vehicle is affected by the quantity of water
inside the ballast tank.
The buoyancy force of the vehicle depends linearly
on the amount of water embarked in the ballast tank.
We tune the position of the ballast tank so that the following symmetry is obtained for the vehicle’s buoyancy: for values of water mass e . 0:5 kg, the vehicle
becomes ‘‘negative’’ (i.e. the gravity force is greater than
the buoyancy force), while for e \ 0:5 kg, the vehicle is
‘‘positive’’ (i.e. the gravity force is less than the buoyancy force).
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Figure 1. Steady-state pitch angle reached by the vehicle as a function of the water contained in the ballast tank positioned on top
of the vehicle’s hull.
Figure 2. Variation in the longitudinal position of the CoG of the vehicle as a function of the water contained in the ballast tank
positioned on top of the vehicle’s hull.
CoG: center of gravity.
As already mentioned before, by introducing a certain mass of water e in the ballast tank at the position
rb , we obtain a shift of the CoG rbg of the entire vehicle.
This is shown in the plot of Figure 2.
representation of the forces acting on the hydrodynamic wing is given in Figure 3.
In this article, we consider the following approximations17 for such induced forces
1 2
rV SCL (a + b)
2
1
D(a + b) = rV2 SCD (a + b)
2
L(a + b) =
Wing characterization
The hydrodynamic wings play a fundamental role to
guarantee the gliding motion of the vehicle. Their contribution affects the equations of the dynamics (2) via
the generalized force t.
For ‘‘numerical convenience,’’ throughout the article, we consider the NACA009 wing profile. The orientation of each wing depends on the angle of attack
(a + b) (see Figure 3), that is, the angle between the
longitudinal direction of the wing xv and the direction
of the velocity vv .
A hydrodynamic wing introduces the lift L(a + b)
and drag D(a + b) forces due to its relative velocity
with respect to the fluid velocity. A schematic
ð5Þ
ð6Þ
where r is the water density, V is the wings’ speed in
the fluid, S is the area of wings’ surface, while the terms
CL and CD are dimensionless coefficients depending on
the chosen wings’ profile and also on the wings’ angle
of attack.
Underwater gliding control problem
We consider the gliding control problem, that is, we
look for a state-feedback control law ue (h, n, e) and an
appropriate wings’ orientation a, affecting the
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Caiti et al.
139
larger gliding angles whenever we have values of water
mass e far from the central value.
We also remark that, in general, the control of the
wings positioned in the tail of the vehicle is not a trivial
task because the system suffers of ‘‘coupling effects.’’ In
fact, any control input on the wings introduces additional moments along the pitch axis, while the differential control strategy introduces moments along the
pitch and the roll axes.
Preliminaries on the path-following problem
Figure 3. Lift and drag forces (top) generated by the
hydrodynamic wings as a function of the angle of attack of the
hydrodynamic wings themselves (bottom). The ‘‘x-axis’’ of the
body-attached reference frame is usually not aligned, see the
angle b, with the direction ‘‘xw ’’ of the relative speed between
the hydrodynamic wing and the fluid.
generalized force as t = t(a), such that a certain
motion is achieved by the vehicle. We notice that setting the wings’ orientation angle a to zero, we achieve
the same maneuverability of a glider with fixed wings.
Hence, we claim that this additional degree of freedom
can be exploited nontrivially, especially if the wings are
independently controlled.
We first notice what happens if the wings’ orientation angle a is fixed to zero. As shown in Figure 4, we
obtain that the equilibrium glide angle reached by the
vehicle is quite ‘‘small’’ if the amount of water contained in the ballast tank is close to its central value,
which is 0.5 kg. On the contrary, the vehicle can achieve
We now define the path-following problem based on
the high-level path-following method presented in
Breivik and Fossen.18 The peculiarity of the method in
Breivik and Fossen18 is the development of control
laws for a generic ideal model of vehicle, according to
the following (backstepping) steps: (a) a point on the
vehicle (‘‘actual particle’’) tracks a desired trajectory
and (b) a control is constructed to achieve the desired
speed computed above.
This strategy is typical of path-following problems
as, unlike trajectory-tracking problems, no time limitations are considered. The actual particle chosen is the
CoG, and the goal here is its convergence to the most
convenient ‘‘path particle,’’ also named ‘‘ideal particle.’’
Here, in our case study, the references are the pitch
and the yaw angles of the velocity of the ideal particle.
We refer and use the control scheme shown in Figure 5.
The model dynamics (2) are implemented in the block
Dynamics. The Reference block solves the step (a) of
the path-following problem described above, in the
sense that it computes the distance between the actual
particle and the ideal particle, together with the state
errors (~
h, n~, ~e). The Feedback block solves the step (b),
namely, it uses such values to generate the statefeedback control actions.
Figure 4. Steady-state glide angle reached by the vehicle as a function of the water contained in the ballast tank positioned on top
of the vehicle’s hull.
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Proc IMechE Part M: J Engineering for the Maritime Environment 228(2)
approximation of it, according to equation (8). For
simplicity, in the control design, we neglect the drag
terms because of about more than one order of magnitude smaller than the lift ones (in the desired operating
conditions). Therefore, we consider an approximate
nonlinearity
! a, V)a
!t = B(b,
Figure 5. Backstepping control scheme. The reference block
computes the reference signals and hence the state errors. The
feedback block implements the control laws to track the
reference signals.
Switching state-feedback velocity control
It is well known that actuating the ballast tank at
‘‘high’’ frequency is particularly inefficient from the
energy consumption point of view. Namely, variations
in the quantity of water e in the tank should be used
only when really needed.
The purpose of the Feedback block is indeed to limit
the chattering behavior of the ballast actuator close to
the equilibrium point. This can be achieved by implementing a switching control law, based on the ‘‘distance’’ from the equilibrium point.
Technically, we propose a feedback control of the
amount of water in the ballast tank that is purely proportional far from the desired path, but null close to
the desired path. We can hence define the control law
"
! T T!
0:1 % sat(Ke ~e) if!(~
e
h , n~ )!4!
ue (~
h, n~, ~e) :¼
ð7Þ
0
otherwise
where Ke 2 R is the feedback gain and e! . 0 is a certain
threshold to be tuned.
Namely, whenever the vehicle is ‘‘close enough’’ to
the desired path, the control of the ballast tank is disabled to avoid the ballast chattering and only the wings
are used to finely reach the desired path.
! a, V) 2 R232 and !t = (t 3 , t6 )T . However, we
with B(b,
remark that in the validation of the control law via
numerical simulations, although the control law is computed neglecting the drag forces, all the terms are considered in the vehicle’s dynamics, including the drag
forces as well.
Regarding the control force !t , we choose these two
components, that is, t3 and t6 because the velocity
angles of pitch and yaw are used here as references. We
now present our switching control law, with the motivation of achieving a ‘‘good’’ convergence to the reference path. In fact, we further remind that close to the
desired path, there is no control action due to the ballast tank in order to avoid an energy-inefficient actuation. As shown later on via numerical simulations, this
design choice is particularly effective in this regard. We
indeed introduce an integral term ‘‘close’’ to the desired
path as follows
8 # $
~
u
>
>
< Kp ~
c
&
#
$
# $
t (~
h, n~, ~e) :¼
Ð ~
~
>
u(t)
u
>
: Kp ~ + Ki
dt
~
c(t)
c
! T T!
e
if!(~
h , n~ )!4!
otherwise
According to the force characterization of the hydrodynamic wings, previously shown in section ‘‘Wing characterization,’’ the generalized force t applied to the
vehicle’s CoG is
ð8Þ
where B(%) is a nonlinear function of the angles
b = (b1 , b2 )T , of the angles a = (a1 , a2 )T and of the
wings’ velocity moduli (V1 , V2 )T = V 2 R2 . The nonlinear function B(%) comes from the forces of lift L(%)
and drag D(%), whose scheme is shown in Figure 3.
The objective here is indeed the choice of a to
recover the desired generalized force t, or a close
ð10Þ
where Kp and Ki 2 R232 are the constant matrix gains
to be tuned.
A possible way to regulate the wing angles a is to
solve an optimization problem online. Given the current angles a0 and b0 , the current velocity moduli V0
and a desired force t& to be reproduced, the angles a
can be chosen as follows
a :¼ arg min s(x)T Qs(x) + (x ! a0 )T R(x ! a0 )
x2R2
subject to : a4x4!
a, Da4x ! a0 4Da
&
! 0 , x, V0 )x
s(x) :¼ t ! B(b
Optimization-based wing control
t = B(b, a, V)a
ð9Þ
ð11Þ
The bounds a, a
! , Da, Da are free design parameters
reflecting the physical limits on the wing actuators. The
weight matrices Q, R are positive definite.
The meaning of the optimization problem (11) is that
we choose an admissible angle a, which is sufficiently
close to its present value a0 , namely, to avoid chattering
on the wings, and also such that the error s between the
induced force and the desired one is sufficiently small.
However, we immediately notice that problem (11) is a
nonlinear optimization problem as s(%) is a nonlinear
function.
Following the approach proposed in Johansen
et al.,19 we can consider a linearized version of problem
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Caiti et al.
141
Figure 6. Three-dimensional plot of the vehicle tracking the (a) desired trajectory and (b) desired helical trajectory and actual
trajectory of a characteristic point of the vehicle.
(11), obtaining the following quadratic problem (QP)
to be solved online
a :¼arg min !s(x)T Q!
s(x) + (x ! a0 )T R(x ! a0 )
x2R2
a, Da4x ! a0 4Da
subject to : a4x4!
s!(x) :¼ t & ! B!ðb0 , a0 , V0 Þa0
&
'
∂ !
ðB(b, a, V)aÞ
!
% (x ! a0 )
∂a
(b0 , a0 , V0 )
ð12Þ
Note that, unlike Johansen et al.,19 the approximation
of the original nonlinearity B(%) in equation (8) with
! in equation (9) is such that B(b,
! a, V) is never sinB(%)
gular. Within this simplifying reduction, there is no
need to address the problem via sequential QPs.
Simulations
We first have performed many numerical simulations
with the goal of tuning the free design parameters introduced in the previous sections. Then, we validate our
proposed switching control strategy. From our numerical experience, we have noticed that in our backstepping control scheme, an undesired chattering behavior
of the ballast actuator would be generated if merely linear control laws are employed.
Our simulation experience indeed confirms the following fact. The action of the ballast tank is more relevant to regulate the pitch variable, while the corrections
determined by the independently controlled hydrodynamic wings play a relevant role in regulating the yaw
variable. Basically, a differential control of the hydrodynamic wings allows to generate roll and yaw motions.
The numerical parameters used in the implemented QP
are a, a = 6158, Da, Da = 658, Q = I and R = 20I.
We present here the gliding motion of the glider, as
shown in Figure 6, that has to follow an helical
trajectory of radius 40 m (see Figure 6 (right)). Such
trajectories are particularly efficient to explore and
sample underwater environments, for instance, sink
holes.20,21 The initial conditions of the vehicle are
h0 = (0, 70, ! 20, 0, 0, 0)T and v0 = (0:05, 0, 0, 0, 0, 0)T .
The initial position error vector is chosen as
h
~ 0 = (0, 30, !20, 0, 0, 0)T . The initial value of the water
contained in the ballast tank is eb = 0:5. The controller
gain has been tuned as
&
'
&
'
!6 0
!0:02
0
Ke = 1, Kp =
, Ki =
0 3
0 0:01
Turning off the ballast tank actuator, when the vehicle is ‘‘close enough’’ to the path, contributes to having
low-energy consumption by correcting trajectory only
through the wings’ regulation. Furthermore, the introduction of the integral term helps to track the desired
path with small state error as shown in Figure 7.
Figure 7 also shows the angular wings’ actuation and
the amount of water in the ballast tank. Figure 7(e)
~
and (f) shows the angular errors ~u and c.
We now show the numerical results obtained if the
control system does not switch to the local control law.
Figure 8 clearly shows that when the vehicles get quite
close to the desired path, the ballast actuation starts
chattering, causing an undesired behavior for the whole
controlled vehicle. We indeed heuristically claim that
switching to a local control law improves the closedloop performance of the vehicle when following a
desired path.
Simulations in the presence of a constant oceanic
current
We now simulate the case in which there is a constant
oceanic current perturbing the dynamics of the
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Figure 7. (a) Time evolution of the position error. Time evolution of the actuator signals: (b) water contained in the ballast tank,
(c) left-wing angle and (d) right-wing angle. Time evolution of the angular errors: (e) pitch and (f) yaw errors.
underwater vehicle. We hence consider a constant force
perturbation t p :¼ (0:1, 0, 0:2, 0, 0, 0)T , whose modulus
is about one-tenth of the one of the hydrodynamic
forces.
In order to obtain good closed-loop performances,
we ‘‘increment’’ the control gains as follows
Ke = 1, Kp =
&
!8
0
'
&
'
0
!0:04
0
, Ki =
4
0 0:02
Figure 9 shows the numerical results of the simulations. The controlled vehicle still performs in a satisfying way, which shows some kind of robustness of the
proposed control algorithm.
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Figure 8. Linear feedback control with no switching. (a) Time evolution of the position error. Time evolution of the actuator
signals: (b) water contained in the ballast tank, (c) left-wing angle and (d) right-wing angle. Time evolution of the angular errors: (e)
pitch and (f) yaw errors.
Conclusion
We have presented a control-oriented model of an
underwater glider with independently controllable
hydrodynamic wings. Through the use of a ballast tank
and the hydrodynamic wings, a simple, backstepping,
control scheme has been implemented to accomplish
some particular 3D path-following maneuvers. We
have used a switching control scheme in order to avoid
the undesired chattering of the ballast tank actuator
when the vehicle gets close to the desired path. The control algorithm is validated via numerical simulations.
We remark that our particular choice of a switching
control law is only justified by simulation heuristics,
not by theoretical arguments.
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Figure 9. The presence of a perturbing oceanic current. (a) Time evolution of the position error. Time evolution of the actuator
signals: (b) water contained in the ballast tank, (c) left-wing angle and (d) right-wing angle. Time evolution of the angular errors:
(e) pitch and (f) yaw errors.
However, the provided example shows the potentialities of using independent actuations for the hydrodynamic wings. In particular, the actuation of the ballast
tank can be limited by just exploiting the corrections of
the wings. This would further increase the energy efficiency of underwater gliders, even if nontrivial maneuvers are to be performed.
Therefore, we expect that this work can motivate
further research studies on the design of more accurate
control schemes exploiting the independent actuation
of the hydrodynamic wings, besides the physical realization of glider prototypes of the kind described here.
From this latter point of view, we finally remark that
the realization of a prototype capable of exploiting its
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hydrodynamic wings together with the waves’ energy
for the surface navigation, and together with the oceanic currents for the underwater navigation, would be
an important breakthrough for the oceanic engineering
community.
Declaration of conflicting interests
The authors declare that there is no conflict of interest.
Funding
This research received no specific grant from any funding agency in the public, commercial or not-for-profit
sectors.
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