Switching control of an underwater glider
with independently controllable wings
Andrea Caiti ∗ , Vincenzo Calabrò ∗ , Stefano Geluardi,
Sergio Grammatico ∗ , Andrea Munafò ∗∗
∗
Department of Energy and Systems Engineering, University of Pisa,
Largo Lucio Lazzarino 1, 56122 Pisa, Italy.
∗∗
Integrated Systems for Marine Environment (ISME),
16126 Genova, Italy.
Abstract: This paper presents a control-oriented dynamic model of an underwater glider with
independently controllable wings. We show that this particular feature is particularly useful to
improve the vehicle’s maneuverability.
The only actuators here used are a ballast tank and two hydrodynamic wings. A switching
control strategy, together with a backstepping control scheme, is designed to limit the action of
the ballast tank and hence to enforce energy-efficient maneuvers.
We consider a case study in which the vehicle has the hydrodynamic wings behind its main hull.
This structure is motivated by the recently-introduced concept of the underwater wave glider,
that is a vehicle capable of both surface and underwater navigation. The control algorithm is
validated via numerical simulations of the vehicle performing three-dimensional path-following
maneuvers.
1. INTRODUCTION
Underwater gliders are winged Autonomous Underwater
Vehicles (AUVs) that, unlike thrusters’ propelled vehicles,
can be employed for long-term and flexible missions, such
as oceanographic sampling and underwater surveillance.
The underwater glider concept (Stommel [1989]) motivated the development of several prototypes, for instance
the “Slocum” (Webb et al. [2001]), the “Spray” (Sherman et al. [2001]) and the “Seaglider” (Eriksen et al.
[2001]). These are all buoyancy-propelled, fixed-winged
gliders which shift internal ballast to control the attitude,
see (Leonard and Graver [2001]).
As underwater glider can be operated in high-efficiency
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 (Leonard and Graver [2001]). Among others,
we refer the interested reader to (Bhatta and Leonard
[2004]), (Mahmoudian and Woolsey [2008]), where some
state-feedback control schemes are proposed for underwater gliders having fixed hydrodynamic wings.
More recently, the need for higher maneuverability motivated the development of underwater gliders with movable
rudder (Noh et al. [2011]) or even with independently
controllable wings (Arima et al. [2008]). For instance, in
(Arima et al. [2009]) many experimental tests are performed with the “Alex” glider to also characterize the
so called “lateral response”. This latter characteristic is
clearly not available if the glider is just equipped with
fixed hydrodynamic wings.
1 Corresponding author Andrea Caiti,
e-mail: a.caiti@dsea.unipi.it.
In this paper, we consider the concept of the underwater
wave glider (Caiti et al. [2012]), capable of both wavepropelled surface navigation and buoyancy-driven underwater motions, in its operation mode of underwater glider.
As a consequence, the hydrodynamic wings of the vehicle
are positioned in the back of the main hull. In particular, we propose a state-feedback switching control scheme
for the three-dimensional path-following task (Encarnaçao
and Pascoal [2000]), (Aguiar and Pascoal [2002]).
The paper is organized as follows. Section 2 contains
the modeling of the vehicle and the characterization of
the actuators, namely the ballast and the independently
controlled wings. Section 3 presents a switching control
scheme for a path-following maneuver. Some simulations
are commented in Section 4. In the last section we conclude
the paper and outline some future lines of research.
2. UNDERWATER GLIDER MODEL
In this section we present the modeling of the underwater glider, with particular attention to the functional
characteristics of the control actuators. The vehicle has
got a time-dependent mass m(t) and a Center of Gravity
(CoG) with variable position rg (t) due to a ballast tank,
positioned on top of the vehicle’s hull.
For any generic vector v, let us 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. The equations governing the dynamics of the
position and of the velocity of the CoG are
Λ + Υ(t)
Υ̇(t)
ṁ(t) b
rgb (t) =
, ṙgb (t) =
−
r (t),
(1)
m(t)
m(t) m(t) g
m
m
kg
m
m
m
kg
kg/s
m
m
m
m
m
m
m
m
Table 1. Numerical parameters of the considered underwater glider
where Λ and Υ(t) are, respectively, the static and the
dynamic contributes of the CoG dynamics.
Considering the mass of water ε(t) contained in the ballast
tank, a standard six-dimensional model for the rigid body
motion (plus the “ballast’s dynamics”) can be obtained as
follows.
η̇ = J (η) v
(2a)
M (ε) v̇ + C (v, ε) v + D (v) v + g (η, ε) +
T (v, ε) ε̇ = τ (2b)
ε̇ = uε
(2c)
where the mass matrix M (ε), the Coriolis matrix C (ε)
already include the added mass effects (Fossen [2002]),
D (v) v is the hydrodynamic drag force acting on vehicle,
g (η, ε) indicates the forces of gravity and buoyancy, and
the term T (v, ε) ε̇ represents the effects of the variable
CoG (1). In (2c) we assume that we can control the velocity
of the CoG ε̇ via the input uε . The generalized force τ
consists on the (controlled) hydrodynamic effects of the
wings, characterized later on.
This modeling technique for vehicles having a variable
position of the CoG has been also used in (Caiti and
Calabrò [2010]) for a hybrid AUV, see (Caffaz et al. [2012])
for further details. In Table 2 the numerical parameters
here used are shown.
2.1 Ballast tank characterization
We assume that the position of the CoG can vary only by
filling and emptying the ballast tank.
Based on the numerical parameters of Table 2, we “characterize” the ballast actuator, positioned on top of the
vehicle’s hull. Therefore, injecting water in the ballast
tank will result in moving the CoG towards 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. As a consequence, also the glide angle of the vehicle
can be regulated.
Buoyancy force of the vehicle depends linearly on the
amount of water embarked in the ballast tank. The posi-
Center of Gravity (CoG) variation [m]
CoG along the Body xéAxis (solid) and zéAxis (dashed)
2
0.15
23.65
−0.02
0
0.1
1
0.1
0.9
0
0.1
0.25
0.10
−0.8
±0.2
0
0.12
0.1
0.08
0.06
0.04
0.02
0
é0.02
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Water contained in the ballast tank [kg]
0.9
1
Fig. 1. Variation of the longitudinal position of the center
of gravity of the vehicle as a function of the water
contained in the ballast tank positionated on top of
the vehicle’s hull.
20
Equilibrium Pitch Angle [deg]
Characteristic Glider parameters
Length
Diameter
Vehicle
Static weight ms
Static CoG position (Λ/ms )x
Static CoG position (Λ/ms )y
Static CoG position (Λ/ms )z
Capacity max b
Ballast Tank
Maximum rate |˙b |max
Position (rbb )x
Position (rbb )y
Position (rbb )z
Width
Hydrodynamic Wings Length
b )
Position (rw
x
b )
Position (rw
y
b )
Position (rw
z
15
10
5
0
é5
é10
é15
é20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Water contained in the ballast tank [kg]
0.9
1
Fig. 2. Steady-state pitch angle reached by the vehicle as
a function of the water contained in the ballast tank
positionated on top of the vehicle’s hull.
tion of the ballast tank has been tuned in such a way that
some symmetry is obtained for the buoyancy: for values of
water mass ε > 0.5 kg the vehicle becomes negative (the
gravity force is greater than the buoyancy force), while for
ε < 0.5 kg the vehicle is positive (the gravity force is less
than the buoyancy force).
By introducing a certain mass of water ε in the ballast
tank at the position rb we obtain a shift of the CoG of the
entire vehicle, as represented in Fig. 1.
2.2 Wings characterization
The hydrodynamic wings play a fundamental role to guarantee the gliding motion of the vehicle. Their contribution
affects the dynamics (2b) via the generalized force τ .
For “numerical convenience”, we consider the NACA009
wing profile.
The orientation of each wing is described by the angle of
attack (α + β), see Figure 3, i.e. the angle between the
longitudinal direction of the wing xω and the direction of
the velocity vω .
A hydrodynamic wing introduces the lift L (α + β) and
drag D (α + β) forces, see Figure 3, due to its relative
velocity with respect to the fluid velocity.
𝜂, 𝑣, 𝜀
𝜂, 𝑣, 𝜀
⋅
𝜏, 𝑢𝜀
𝜂, 𝑣 , 𝜀
Fig. 3. Lift and drag forces generated by the hydrodynamic
wings as a function of the angle of attack of the
hydrodynamic wings themselves.
In this paper we consider the following approximations
(Chwang and Wu [1975]) for such induced forces:
1
L(α + β) = ρV 2 SCL (α + β)
(3)
2
1
(4)
D(α + β) = ρV 2 SCD (α + β)
2
where ρ is water density, V is the wings speed in the fluid,
S is the are of wings surface, while the terms CL and
CD are dimensionless coefficient depending on the chosen
wings profile and also on the wings angle of attack.
3. UNDERWATER GLIDING CONTROL PROBLEM
The gliding control problem we address here is the problem
of finding a state-feedback control law uε (η, ν, ε) and the
wings orientation α, affecting τ = τ (α), such that a certain
motion is achieved by the vehicle.
We notice that setting the wings orientation angle α to
zero, we achieve the same maneuverability of a glider with
fixed wings. Hence we claim that this additional degree of
freedom can be actually exploited, especially if the wings
are independently controlled.
Now, let us also notice what happens if the wings orientation angle α is fixed to zero. In this case, we have that,
see Figure 5, 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, that is 0.5
kg. On the contrary, the vehicle can achieve larger gliding
angles whenever we have values of water mass ε far from
the central value.
Note 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.
3.1 Preliminaries on the path-following problem
The high-level control strategy used in this work is based
on the path-following method (Breivik and Fossen [2005]).
We indeed refer to this reference for the details on the
definition of the path-following problem.
Fig. 4. Backstepping control scheme. The feedforward
block computes the reference signals and hence the
state errors. The feedback block implements the control laws to track the reference signals.
The peculiarity of the method in (Breivik and Fossen
[2005]) 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; (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”.
In our case, the references are the pitch and the yaw
angles of the velocity of the ideal particle. The control
scheme is presented in Fig.4. The model dynamics (2b)
are implemented in the block Dynamics.
The Feedforward block solves the step (a) of the pathfollowing problem described above, in the sense that is
computes the distance between the actual particle and the
ideal particle, together with the state errors (η̃, ν̃, ε̃). The
Feedback block solves the step (b), namely it uses such
values to generate the state-feedback control actions.
3.2 Switching state-feedback velocity control
In the Feedback block, a switching control is implemented
in order to limit the chattering behavior of the ballast
actuator close to the equilibrium point. The latter phenomenon would be definitively undesired because, from
the practical point of view, actuating the ballast tank
at “high” frequency is particularly inefficient from the
energy-consumption point of view.
Technically, we propose a feedback control of the amount
of water in the ballast tank that is purely proportional far
from the desired path:
0.1 · sat(Kε ε̃) if (η̃ > , ν̃ > ) ≤ ē
uε (η̃, ν̃, ε̃) =
(5)
0
otherwise
A possible way to regulate the wings angles α is to solve
an optimization problem online. Given the current angles
α0 , β0 , the current velocity moduluses V0 and a desired
force τ ∗ to be reproduced, the angles α can be chosen as
follows.
α := arg min2 s(x)> Qs(x) + (x − α0 )> R(x − α0 )
80
Equilibrium Glide Angle [ deg ]
60
40
20
x∈R
0
subject to:
α ≤ x ≤ α, ∆α ≤ x − α0 ≤ ∆α
where:
s(x) := τ ∗ − B̄ (β0 , x, V0 ) x.
ï20
ï40
(9)
ï60
ï80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Water contained in the ballast tank [ kg ]
0.8
0.9
1
Fig. 5. Steady-state glide angle reached by the vehicle as
a function of the water contained in the ballast tank
positionated on top of the vehicle’s hull.
where Kε ∈ R is the feedback gain and ē ∈ R+ 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.
3.3 Optimization-based wings control
According to the force characterization of the hydrodynamic wings, see Section 2.2, the generalized force τ applied to the vehicle’s CoG is
τ = B (β, α, V ) α,
(6)
where B(·) is a nonlinear function of the angles β =
(β1 , β2 )> , of the angles α = (α1 , α2 )> and of the wingsvelocities moduluses (V1 , V2 )> = V ∈ R2 . The nonlinear
function B(·) comes from the forces of lift L(·) and drag
D(·) forces that are shown in Figure 3.
The objective here is indeed the choice of α to recover the
desired generalized force τ , or a close approximation of it,
according to equation (6).
For simplicity, in the control synthesis we neglect the drag
terms because of about one order of magnitude smaller
than the lift ones (in the desired operating conditions).
Therefore, we consider an approximate nonlinearity:
τ̄ = B̄ (β, α, V ) α
(7)
with B̄ (β, α, V ) ∈ R2×2 and τ̄ = (τ3 , τ6 )> .
We choose this two components because the velocity angles
of pitch and yaw are here used as references.
In order to obtain a good convergence, an integral term is
introduced “close” to the desired path:
τ ∗ (η̃, ν̃, ε̃) =


 Kp · ψ̃θ̃
Z θ̃(t)

dt
 Kp · ψ̃θ̃ + Ki
ψ̃(t)
if (η̃ > , ν̃ > ) ≤ ē
(8)
otherwise
where Kp and Ki ∈ R2×2 are constant matrix gains to be
tuned.
The bounds α, α, ∆α, ∆α are free design parameters
reflecting the physical limits on the wings actuators. The
weight matrices Q, R are positive definite. The problem (9)
is a nonlinear optimization problem as s(x) is a nonlinear
function of x.
Following the approach proposed in (Johansen et al.
[2004]), we consider a linearized version of problem (9)
leading the following Quadratic Problem (QP) to be solved
online.
α := arg min2 s̄(x)> Qs̄(x) + (x − α0 )> R(x − αo )
x∈R
subject to:
α ≤ x ≤ α, ∆α ≤ x − α0 ≤ ∆α
where:
s̄(x) :=τ ∗ − B̄ (β0 , α0 , V0 )α0
∂
B̄(β, α, V )α
· (x − α0 ).
−
∂α
(β0 ,α0 ,V0 )
(10)
Note that, unlike (Johansen et al. [2004]), the approximation of the original nonlinearity B(·) in (6) with B̄(·)
in (7) is such that B̄(β, α, V ) is never singular. Within
this simplifying reduction, there is no need to address the
problem via sequential QPs.
4. SIMULATIONS
Many numerical simulations have been performed in order
to tune the free design parameters and also to validate
the proposed switching-control strategy. We noticed that
in our backstepping control scheme, an undesired chattering behavior of the ballast actuator would be generated
without the use of the proposed switching strategy.
Our simulation experience confirms what basic intuition
would suggest: 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 α, α = ±15 deg, ∆α, ∆α = ±5 deg, Q = I, R = 20I.
We present here the gliding motion of the glider, shown
in Figure 6, that has to follow an helical trajectory of
radius 40 m, see Figure 7. Such trajectories are particularly efficient to explore and sample underwater sink holes
(Andonian et al. [2010a,b]). The initial vehicle states are
η0 = (0, 70, −20, 0, 0, 0)> and v0 = (0.05, 0, 0, 0, 0, 0)> .
Position error
40
35
é213.4
30
é213.6
25
6P
é213.8
éDown
é214
15
é214.4
10
é214.6
é214.8
5
é215
29.5
26.5
26
31
25.5
100
200
300
Time [ s ]
400
500
600
North
East
Fig. 8. Position error over time.
Fig. 6. Three-dimensional plot of the vehicle tracking the
desired trajectory.
Path following
Actual
Desired
Water in the ballast tank [ kg ]
27
0
0
30
30.5
é215.2
100
Ballast tank
1
0.5
0
0
50
100
150
200
250
300
350
400
450
500
300
350
400
450
500
300
350
400
450
500
Left wing
_ [ deg ]
100
0
ïz [ m ]
20
é214.2
ï100
50
0
ï50
ï100
0
ï200
50
100
150
200
20
0
ï20
x[m]
ï40
ï40
ï20
0
20
40
60
80
_ [ deg ]
100
ï300
40
50
0
ï50
ï100
y[m]
Fig. 7. Desired helical trajectory and actual trajectory of
a characteristic point of the vehicle.
250
Right wing
0
50
100
150
200
250
Time [ s ]
Fig. 9. Actuator signals over time.
5. CONCLUSION
The initial position error vector is chosen as η̃0 =
(0, 30, −20, 0, 0, 0)> . The initial value of the water contained in the ballast tank is εb = 0.5. The controller gain
have been tuned as
−6 0
−0.02
0
Kε = 1, Kp =
, Ki =
.
0 3
0 0.01
Turning off the ballast tank actuator, when the vehicle
is “close enough” to the path, contributes in having low
energy-consumption by correcting trajectory only through
the wings control. Furthermore, the introduction of the
integral term helps to track the desired path with small
state error as shown in Figure 8. Figure 9 shows the
angular wings actuation and the amount of water in the
ballast tank. Figure 10 shows the angular errors θ̃ and ψ̃.
In this paper we have presented a control-oriented model
of an underwater glider with independently controllable
hydrodynamic wings. Through the use of a ballast tank
and of the hydrodynamic wings, a simple, backstepping,
control scheme has been presented to accomplish some particular three-dimensional path-following maneuvers. We
have used a switching control scheme to avoid the undesired chattering of the ballast tank actuator whenever
the vehicle gets close to the desired path. The control
algorithm is validated via numerical simulations. We remark that our switching control law is only justified by
simulations heuristics, not by theoretical analysis.
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
Pitch angle error
6e [ deg ]
100
50
0
ï50
ï100
0
100
200
300
500
600
700
800
500
600
700
800
Yaw angle error
200
6s [ deg ]
400
100
0
ï100
0
100
200
300
400
Time [ s ]
Fig. 10. Angular errors over time.
further increase the energy efficiency of underwater gliders,
even if non-trivial 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 here described.
From this latter point of view, it would be remarkable
the realization of a prototype capable of exploiting its
hydrodynamic wings together with the waves energy for
the surface navigation, and together with the oceanic
currents for the underwater navigation.
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