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Saturation based nonlinear depth and yaw control of underwater
vehicles with stability analysis and real-time experiments
�
E. Campos b , c , d , ∗, A. Chemori c , V. Creuze
c , J. Torres a , b , R. Lozano
b
a Automatic Control Department, CINVESTAV, México D.F., México b UMI-LAFMIA,CINVESTAV-CNRS, México, D.F., México c LIRMM, CNRS-Université Montpellier 2, Montpellier, France d CONACYT-Universidad del Istmo, Tehuantepec, Oaxaca, México
a r t i c l e i n f o
Article history:
Received 19 May 2015
Revised 14 March 2017
Accepted 5 May 2017
Keywords:
Underwater vehicle
Nonlinear PD and PD+ controller
Saturation
Real-time experiments
a b s t r a c t
This paper deals with two nonlinear controllers based on saturation functions with varying parameters,
for set-point regulation and trajectory tracking on an Underwater Vehicle. The proposed controllers com-
bine the advantages of robust control and easy tuning in real applications. The stability of the closed-loop
system with the proposed nonlinear controllers is proven by Lyapunov arguments. Experimental results
for the trajectory tracking control in 2 degrees of freedom, these are the depth and yaw motion of an
underwater vehicle, show the performance of the proposed control strategy.
The restoring forces and moments are generated by the weight
W
and the buoyancy force f B , this latter, always acts in the oppo-
ite direction of vehicle weight, that is:
f B = −[
0
0
B
]
f W
=
[
0
0
W
]
(8)
here B represents the magnitude of the buoyancy force, defined
ccording to the Archimedes’ principle; W = mg is the vehicle’s
eight, with g the gravitational acceleration. Notice that these
orces are defined with respect to the earth-fixed-frame. Now, us-
ng the zyx -convention for navigation and control applications [5] ,
he transformation matrix J 1 (η2 ) = R z,ψ
R y,θ R x,φ is introduced in or-
er to obtain the buoyancy force and weight with respect to the
ody-fixed-frame:
B = J 1 (η2 ) −1 f B , F W
= J 1 (η2 ) −1 f W
(9)
hen, the restoring forces acting on the vehicle are f g = F B + F W
,
eading to:
f g =
[
(B − W ) sin (θ ) (W − B ) cos (θ ) sin (φ) (W − B ) cos (θ ) cos (φ)
]
(10)
n the other hand, the restoring moments depend on the positions
f the center of gravity (CG) and the center of buoyancy (CB), as
e can notice in the following equation:
g = r w
× F W
+ r b × F B (11)
here r w
= [ x w
, y w
, z w
] T and r b = [ x b , y b , z b ] T represent the posi-
ions of the center of gravity and the center of buoyancy, respec-
ively. In our case the origin of the body-fixed-frame is chosen in
he center of gravity, this implies that r w
= [0 , 0 , 0] T , while the
enter of buoyancy is r b = [0 , 0 , −z b ] T . For practical purposes, the
uoyancy force is greater than the weight, i.e. B − W = f > 0.
b
52 E. Campos et al. / Mechatronics 45 (2017) 49–59
Fig. 4. Saturation function with fixed parameters.
w
I
t
c
t
τ
w
σ
σ
w
p
n
τ
T
τ
w
b
d
u
k
d
u
Then, from Eqs. (10) and (11) , we obtain the vector of restoring
forces and moments as follows:
g(η) =
[f g
m g
]=
⎡
⎢ ⎢ ⎢ ⎢ ⎣
f b sin (θ ) − f b cos (θ ) sin (φ) − f b cos (θ ) cos (φ)
−z b B cos (θ ) sin (φ) −z b B sin (θ )
0
⎤
⎥ ⎥ ⎥ ⎥ ⎦
(12)
2.2.3. Control inputs: forces and torques generated by the thrusters
The forces generated by the thrusters T 1 to T 6 are denoted
f 1 to f 6 , and are defined by: f 1 = [0 , 0 , f 1 ] T , f 2 = [0 , 0 , f 2 ]
T , f 3 =[0 , 0 , f 3 ]
T , f 4 = [ f 4 , 0 , 0] T , f 5 = [ f 5 , 0 , 0] T , f 6 = [0 , f 6 , 0] T , as illus-
trated in Fig. 2 . Then, the translation motions are produced by:
τ1 =
[
τX
τY
τZ
]
=
[
f 4 + f 5 f 6
f 1 + f 2 + f 3
]
(13)
and the torques generated by the above forces, are defined as fol-
lows:
τ2 =
6 ∑
i =1
l i × f i (14)
where l i = (l ix , l iy , l iz ) is the position vector describing where the f i (for i = 1 , .., 6 . ) forces apply, with respect to the body-fixed refer-
ence frame. The torques generated by the thrusters are then de-
scribed by:
τ2 =
[
τK
τM
τN
]
=
[
l 2 y f 2 + l 3 y f 3 l 2 x f 2 + l 3 x f 3 + l 1 x f 1
l 4 y f 4 + l 5 y f 5
]
(15)
Finally, the vector of control inputs is expressed as follows:
τ =
⎡
⎢ ⎢ ⎢ ⎢ ⎣
f 4 + f 5 f 6
f 1 + f 2 + f 3 l 2 y f 2 + l 3 y f 3
l 2 x f 2 + l 3 x f 3 + l 1 x f 1 l 4 y f 4 + l 5 y f 5
⎤
⎥ ⎥ ⎥ ⎥ ⎦
(16)
3. Proposed control strategy
In this section, nonlinear PD and PD+ controllers based on sat-
uration functions with variable parameters are introduced. Both of
them are proposed for set point regulation as well as for trajectory
tracking control. The stability analysis of the resulting closed-loop
system for both cases is detailed.
3.1. Nonlinear PD controller with gravity and buoyancy compensation
Considering the dynamics given by Eqs. (1) and (2) , the PD con-
trol law with static feedback gains and gravity/buoyancy compen-
sation is given by:
τ = g(η) − J T (η) τPD (17)
with
τPD = K p e (t) + K d
de (t)
dt (18)
where K p , K d ∈ R
6 ×6 are diagonal, positive definite matrices, and
e (t) = η − ηd represents the error.
In order to improve the performance of the closed-loop system,
we propose to introduce (in each term of Eq. (18) ) a saturation
function σb (h ) illustrated in Fig. 4 and defined by:
σb (h ) =
⎧ ⎨
⎩
b i f h > b
h i f | h |≤ b
−b i f h < −b
(19)
here b is a positive constant, and h represents a linear function.
n our case, the terms to which this saturation will be applied are
he error and its time derivative.
Then, if we introduce the above saturation function into in the
ontrol law (18) , we obtain the following nonlinear PD (NLPD) con-
roller:
NLPD = σb p
[ K p e (t)] + σb d
[K d
de (t)
dt
](20)
here
b p [ K p e (t)] =
⎡
⎢ ⎢ ⎣
u p1 0 . . . 0
0 u p2 . . . 0
. . . . . .
. . . . . .
0 0 . . . u pn
⎤
⎥ ⎥ ⎦
(21)
b d
[K d
de (t)
dt
]=
⎡
⎢ ⎢ ⎣
u d1 0 . . . 0
0 u d2 . . . 0
. . . . . .
. . . . . .
0 0 . . . u dn
⎤
⎥ ⎥ ⎦
(22)
ith u pj = σb p j
[ k pj e j (t)] ; u dj = σb dj
[ k dj de j (t)
dt ] ; where k pj , k dj are
ositive constants, for all j = 1 . . . n.
Without loss of generality, let us consider now the scalar case,
amely:
NLPD 1 = σb p1
[ k p1 e 1 (t)] + σb d1
[k d1
de 1 (t)
dt
](23)
he above equation can be rewritten in a compact form as follows:
NLPD 1 =
2 ∑
i =1
u i (24)
here u i = σb i (k i h i ) represents the saturation function, with b 1 =
¯ p1 , b 2 = b d1 , k 1 = k p1 , k 2 = k d1 ; h 1 is the error and h 2 its first
erivative. Then, from Eq. (19) u i can be rewritten as:
i =
⎧ ⎨
⎩
b i i f k i h i > b i k i h i i f | k i h i | ≤ b i −b i i f k i h i < −b i
(25)
In the above equation, we can notice that the linear function
i h i is saturated by | h i | = b i /k i . At that time, we define:
i := b i /k i (26)
Then, we can rewrite Eq. (25) as follows:
i =
{sign (h i ) b i i f | h i | > d i
b i d −1 i
h i i f | h i | ≤ d i (27)
E. Campos et al. / Mechatronics 45 (2017) 49–59 53
Fig. 5. Saturation function with various values of parameter μ.
w
1
s
w
s
a
a
u
τ
w
k
k
a
s
m
e
(
b
a
b
w
u
∀
r
f
τ
w
k
k
e
t
o
l
s
p
T
P
τ
w
t
K
K
t
b
a
P
η
(
τ
a
t
t
M
a
M
t
M
[N
l
t
t
V
here the tuning parameters of the controller are b i and d i , ∀ i = , 2 . Moreover, considering that we have:
ign (h i ) b i = h i sign (h i ) b i h
−1 i
(28)
hich can be simplified as:
ign (h i ) b i = | h i | b i h
−1 i
(29)
nd considering that | h i | h −1 i
= | h i | −1 h i , Eq. (27) can be rewritten
s follows:
i =
{b i | h i | −1 h i i f | h i | > d i b i d
−1 i
h i i f | h i | ≤ d i (30)
Consequently, the control law (23) can be rewritten as:
NLPD 1 = u 1 + u 2 = k p1 (·) e 1 (t) + k d1 (·) e 1 (t) (31)
ith:
p1 (·) =
{b p1 | e 1 (t) | −1 i f | e 1 (t) | > d p1
b p1 d −1 p1
i f | e 1 (t) | ≤ d p1 (32)
d1 (·) =
{b d1 | e 1 (t) | −1 i f | e 1 (t) | > d d1
b d1 d −1 d1
i f | e 1 (t) | ≤ d d1
(33)
The advantage of this formulation is that the forces and torques
re limited by the parameters b p1 and b d1 . Consequently, we are
ure of the boundedness of the control input. However, some cases
ay require slightly larger forces and torques to correct the system
rrors, that is why we propose that the saturation value b i in Eq.
30) should be changed as follows:
¯ i = b i | h i | μi i f | h i | > d i (34)
nd
¯ i = b i | d i | μi i f | h i | ≤ d i (35)
ith b i a positive constant, and μi ∈ [0, 1].
Now, introducing Eqs. (34) and (35) into (30) , we obtain:
i =
{b i | h i | μi | h i | −1 h i i f | h i | > d i b i | d i | μi d −1
i h i i f | h i | ≤ d i
(36)
i = 1 , 2 and μi ∈ [0, 1].
The plots of the above function for different values of the pa-
ameter μi are shown in Fig. 5 .
Consequently, the nonlinear PD control law based on saturation
unction with variable parameters can be expressed as:
NLPD j = k p j (·) e j (t) + k dj (·) e j (t) (37)
ith:
p j (·) =
{b p j | e j (t) | (μpj −1) i f | e j (t) | > d p j
b p j d (μpj −1)
p j i f | e j (t) | ≤ d p j
(38)
dj (·) =
{b dj | e j (t) | (μdj −1) i f | e j (t) | > d dj
b dj d (μdj −1)
dj i f | e j (t) | ≤ d dj
(39)
∀ μpj , μdj ∈ [0, 1]
From Fig. 5 , it can be noticed that if μpj = μdj = 1 , the nonlin-
ar PD controller given by (37) degenerates into the linear PD con-
roller given by (18) . Besides, if μpj = μdj = 0 , we obtain the case
f a constant saturation. To summarize, we can conclude that the
inear PD controller and the nonlinear PD controller with a simple
aturation function, defined by Eq. (19) , are particular cases of the
roposed controller.
heorem 1. For the case of set-point regulation, under the nonlinear
D control (NLPD) with gravity compensation
= g(η) − J T (η) [ K p (·) e + K d (·) e ] (40)
here the feedback gains K p ( ·) and K d (·) have the following struc-
ure:
p (·) =
⎡
⎢ ⎢ ⎣
k p1 (·) 0 . . . 0
0 k p2 (·) . . . 0
. . . . . .
. . . . . .
0 0 . . . k pn (·)
⎤
⎥ ⎥ ⎦
> 0 (41)
d (·) =
⎡
⎢ ⎢ ⎣
k d1 (·) 0 . . . 0
0 k d2 (·) . . . 0
. . . . . .
. . . . . .
0 0 . . . k dn (·)
⎤
⎥ ⎥ ⎦
> 0 (42)
he system (1) is asymptotically stable if k pj ( ·) and k dj ( ·) are defined
y (38) and (39) respectively and if the underwater vehicle is moving
t low speed.
roof. In the case of set-point regulation ηd is constant, then
˙ d = 0 and
˙ e = ˙ η. As a consequence the control law given by Eq.
40) can be rewritten as:
= g(η) − J T (η)[ K p (·) e + K d (·) η] (43)
In what follows we will suppose that θ � = ±π /2, in order to
void possible singularities of J( η) matrix, see [5] . Now assuming
hat w e = 0 , the injection of the control law (43) into (1) , leads to
he following closed-loop system:
ν + C(ν) ν + D (ν) ν = −J T (η)[ K p (·) e + K d (·) η] (44)
nd if we consider the transformation (2) , we obtain:
ν + C(ν) ν + D (ν) ν = −J T (η)[ K p (·) e + K d (·) J(η) ν] (45)
Let us define K dd (·) = J T (η) K d (·) J(η) , then the previous equa-
ion can be rewritten as:
ν + C(ν) ν + D (ν) ν = −J T (η) K p (·) e − K dd (·) ν (46)
The closed-loop system (46) can be represented as
d
dt
[e ν
]=
J(η) νM
−1 [ −J T (η) K p (·) e − K dd (·) ν − C(ν) ν − D (ν) ν]
](47)
otice that the origin of the state space model is a unique equi-
ibrium point. Now, in order to proof the asymptotic stability of
he closed-loop system we propose the following Lyapunov func-
ion candidate:
( e , ν) =
1
2
νT Mν +
∫ e
0
ξ T K p (ξ ) dξ (48)
54 E. Campos et al. / Mechatronics 45 (2017) 49–59
V
V
V
fi
M
T
l
τ
w
K
K
a
b
a
P
l
M
w
[
w
a
t
f
V
a
w
b
V
N
Z
s
D
V
s
n
where ∫ e 0 ξ
T K p (ξ ) dξ =
∫ e 1 0
ξ1 k p1 (ξ1 ) dξ1 +
∫ e 2 0
ξ2 k p2 (ξ2 ) dξ2 + ∫ e 3 0
ξ3 k p3 (ξ3 ) dξ3 + . . . +
∫ e n 0 ξn k pn (ξn ) dξn .
Now, considering that the inequality
e j k p j (·) ≥ α j (| e j | ) (49)
is satisfied with the class K functions
α j (| e j | ) =
⎧ ⎪ ⎨
⎪ ⎩
b j | e j | μpj e j
a + | e j | i f | e j | > d j
b j d μpj
j e j
a + d j i f | e j | ≤ d j
(50)
with b pj > b j , a > 0 and d pj < d j . Then, according to Lemma 2 from
[8] one deduces the following: ∫ e
0 ξ T K p (ξ ) dξ > 0 ∀ e � = 0 ∈ R
n (51)
and ∫ e
0 ξ T K p (ξ ) dξ → ∞ as ‖ e ‖→ ∞ (52)
Therefore, the Lyapunov function candidate V ( e, ν) is posi-
tive definite and radially unbounded. Now, for underwater vehi-
cles move at low speed the time differentiation of (48) , along
the trajectories of ν and e , is done with the assumptions that is
M = M
T > 0 , C ( ν) is skew symmetrics and D ( ν) is definite positive,
more details see [3] . Then, using the Leibniz’ rule for differentia-
tion of integrals, the time derivative of the Lyapunov function can-
didate is:
˙ ( e , ν) = νT M ν + e T K p (e ) J(η) ν (53)
by substituting the closed-loop Eq. (46) into (53) one obtains:
˙ ( e , ν) = −νT J T (η) K p (e ) e − νT K dd (η, ˙ e ) ν
−νT C(ν) ν − νT D (ν) ν + e T K p (e ) J(η) ν (54)
since K p (e ) = K
T p (e ) and C(ν) = −C T (ν) , Eq. (54) becomes:
˙ ( e , ν) = −νT [ K dd (η, ˙ e ) + D (ν)] ν (55)
Recall that K d = K
T d
> 0 , therefore K dd = K
T dd
> 0 , and assuming
that D( ν) > 0, then one can conclude that ˙ V ( e , ν) is negative semi-
definite. Therefore the stability of the equilibrium point is guar-
anteed. In order to prove the asymptotic stability, the Krasovskii–
LaSalle’s theorem can be used, let
� =
{[e ν
]: ˙ V ( e , ν) = 0
}=
{[e ν
]=
[e 0
]∈ R
2 n
}(56)
introducing ν = 0 and
˙ ν = 0 into Eq. (46) leads to the unique in-
variant point e = 0 . Therefore, we conclude that equilibrium point
is asymptotically stable. � �
3.2. Nonlinear PD+ controller
For the case of trajectory tracking problem, we propose to use
a nonlinear PD+ controller with the same feedback gains as the
previous controller.
Based on Eq. (2) , the following kinematic transformations can
be obtained (see [5] for more details):
η = J(η) ν +
˙ J (η) ν �⇒
˙ ν = J −1 (η) [ η − ˙ J (η) J −1 (η) η]
Applying the previous transformations to the dynamic model
56 E. Campos et al. / Mechatronics 45 (2017) 49–59
Table 3
Parameters of the controller for case 1.
Depth b p3 = 70 d p3 = ∞ μp3 = 1
b d3 = 5 d d3 = ∞ μd3 = 1
Yaw b p6 = 11 d p6 = ∞ μp6 = 1
b d6 = 1 . 5 d d6 = ∞ μd6 = 1
Table 4
Parameters of the controller for case 2.
Depth b p3 = 20 d p3 = 0 . 05 μp3 = 0 . 1
b d3 = 13 d d3 = 0 . 25 μd3 = 0 . 2
Yaw b p6 = 4 d p6 = 5 . 72 μp6 = 0 . 09
b d6 = 5 d d6 = 14 . 32 μd6 = 0 . 2
Fig. 9. L2ROV with the added two buoyant floats and a rigid plastic sheet, which
will increase the buoyancy force and damping along z axis.
Table 5
Evaluation criteria for scenario 1.
RMSE z (m) INT z RMSE ψ (deg) INT ψ
Case 1 0.0087 4657 0.04 507.2
Case 2 0.0044 4913 0.03 647.6
Table 6
Evaluation criteria for scenario 2.
RMSE z (m) INT z RMSE ψ (deg) INT ψ
Case 1 0.1106 19,356 0.0146 611.19
Case 2 0.0605 19,831 0.0060 721.36
t
I
w
a
R
t
0
4
t
d
e
e
c
u
4
p
t
2
a
i
o
F
i
r
o
t
l
τ
o
f
e
t
v
t
c
f
c
(ISTSE) presented in [23] . In the second step the gains have been
manually adjusted to get best results. The obtained parameters are
summarized in Table 3 .
The control parameters for the case 2 are given in Table 4 , they
are obtained by a heuristic method based on the following steps:
• First d pj is chosen, taking into account that the interval
[ −d pj , d pj ] is the linear region of the proposed controller.
• Considering b dj = 0 and μpj = 0 ; b pj is increased until the
closed-loop system oscillates.
• d dj is chosen bigger than d pj , and μdj = 0 .
• Then b dj is increased until the system oscillations decrease.
• Finally, μpj and μdj are adjusted to improve the system behav-
ior, considering μpj < μdj .
Fig. 10 −(a ) shows the obtained results for trajectory tracking in
depth, the corresponding tracking error, and the control input for
the controller defined in Case 1. Fig. 10 −(b) shows the evolution
of the tracking in yaw, the corresponding tracking error, and the
control input produced by the thrusters.
Fig. 11 −(a ) depicts the experimental results for trajectory track-
ing in depth, the corresponding tracking error, and the control in-
put for the controller defined in Case 2. Fig. 11 −(b) shows the evo-
lution of the tracking in yaw, the corresponding tracking error and
the yaw control input. Moreover, we can observe that the yaw mo-
tion converges to the desired trajectory in less than 1.5 s.
In order to evaluate the tracking performance of the proposed
controllers, let us compute the Root Mean Square Error (RMSE) for
z and ψ . In addition, the integral of control inputs (the applied
force and torque) are computed to estimated the energy consump-
ion used in each case, that is:
NT =
∫ t 2
t 1
| τ (t) | dt (67)
here t 1 = 2 s, since in this time for both cases the system’s states
re close to their desired values, and t 2 = 30 s.
From the results of Table 5 , we observe that the RMSE z and
MSE ψ
of case 2 are smaller than in case 1. It can be observed
hat steady-state errors z and ψ are approximately 8.7 mm and
.04 deg for the case 1, while for the case 2 are approximately
.4 mm and 0.03 deg respectively. Moreover, notice that the quo-
ients between INT z and INT ψ
from case 1 and 2 are:
4913 4657
= 1 . 0550
647 . 6 507 . 02
= 1 . 27
(68)
This means that energy consumption for trajectory tracking in
epth, using the controller defined in Case 2, is 1.055 times the
nergy consumption using the controller defined in Case 1. While
nergy consumption for trajectory tracking in heading, using the
ontroller defined in Case 2, is 1.27 times the energy consumption
sing the controller defined in Case 1.
.3. Scenario 2: robustness test
The main goal of this scenario is to test the robustness of the
roposed controllers towards uncertainties in the parameters of
he model. During the real-time experiments, we have added two
00 g buoyant floats (as illustrated in Fig. 9 ), increasing the buoy-
ncy of 330%, and a large (45 cm × 10 cm) rigid plastic sheet (as
llustrated in Fig. 9 ), increasing the rotational damping along z axis
f about 90%.
The obtained experimental results for case 1 are shown in
ig. 12 . The tracking performance of the control system for depth
s degraded. Indeed, depth control of the vehicle was not able to
each the desired trajectory. We can notice that steady-state error
n z and ψ are approximately 11 cm and 0.01 deg, respectively. For
he yaw motion, the vehicle converges to the desired trajectory in
ess than 1 s (as noticed in Fig. 12 −(a ) ). The force τ z and torque
ψ
generated by the thrusters are displayed at the bottom curves
f Fig. 12 .
Fig. 13 shows that the performance of the system is less af-
ected in case 2. Indeed, it can be observed that the steady-state
rrors on z and ψ are approximately 6 cm and 0.006 deg, respec-
ively. The yaw motion is the less affected, since the vehicle con-
erge to the desired trajectory in less than 1 s. The generated con-
rol input (force τ z and torque τ p si ) are displayed in the bottom
urves of Fig. 13 .
Now, the RMSE and the integral of the applied force and torque
or both cases are summarized in Table 6 :
According to Table 6 the quotients between INT z and INT ψ
from
ase 1 and 2 are:
19831 = 1 . 0245
721 . 36 = 1 . 1803 (69)
19356 611 . 19
E. Campos et al. / Mechatronics 45 (2017) 49–59 57
Fig. 10. Experimental results of scenario 1 in the case 1: trajectory tracking of depth and yaw versus time, error signal of both motions and evolution of the control inputs
generated by the thrusters.
Fig. 11. Experimental results of scenario 1 in the case 2: trajectory tracking of depth and yaw versus time, error signal of both motions and evolution of the control inputs
generated by the thrusters.
58 E. Campos et al. / Mechatronics 45 (2017) 49–59
Fig. 12. Experimental results of scenario 2 in the case 1: trajectory tracking of depth and yaw versus time, error signal of both motions and evolution of the control inputs
generated by the thrusters.
Fig. 13. Experimental results of scenario 2 in the case 2: trajectory tracking of depth and yaw versus time, error signal of both motions and evolution of the control inputs
generated by the thrusters.
t
t
p
c
b
This means that energy consumption for trajectory tracking in
depth, using the controller defined in Case 2, is 1.05 times the
energy consumption using the controller define in Case 1. While
energy consumption for trajectory tracking in heading, using the
controller defined in Case 2, is 1.27 times the energy consump-
ion using the controller defined in Case 1. We can observe that
he quotients obtained in this scenario are very similar as in the
revious scenario, see Eq. (68) . Moreover, one can notice that the
losed-loop system with the nonlinear PD+ controller, represented
y case 2, is less affected. It can be observed that steady-state er-
E. Campos et al. / Mechatronics 45 (2017) 49–59 59
r
1
0
i
c
v
c
t
5
p
s
s
d
j
d
t
w
b
t
s
A
o
h
(
E
m
R
[
[
[
t
u
o
2
T
C
p
p
2
j
ors z and ψ are approximately 11 cm and 0.01 deg for the case
, while for the case 2 their values are approximately 6 cm and
.006 deg respectively. Moreover, notice that the chattering is large
n the second case than in the first one. This is due to the stronger
ompromise between performance and robustness imposed by the
ariable saturation case. Then, we can conclude that the proposed
ontrol strategy demonstrated a good ability to deal with parame-
ers’ uncertainties.
. Conclusion and future work
In this paper, a nonlinear PD and PD+ controllers have been
roposed for depth and yaw control of underwater vehicles. The
tability analysis for the resulting closed-loop system for both
et-point regulation and trajectory tracking control has been ad-
ressed. The proposed controllers have been implemented for tra-
ectory tracking in depth and yaw motions with the L2ROV un-
erwater vehicle. The obtained experimental results demonstrate
he effectiveness and the robustness of the proposed controllers to-
ards uncertainties on the parameters of the system (damping and
uoyancy changes). The future work will consist in implementing
he integral term of the controller in order to improve the steady-
tate performance of the closed-loop system.
cknowledgements
This work was supported by the PCP research project, in collab-
ration with the Tecnalia foundation. The L2ROV underwater ve-
icle has been funded by the Region Languedoc-Roussillon council
ARPE MiniROV). The authors greatly acknowledge support of the
uropean Union through FEDER grant no. 49793 for the develop-
ent of the Leonard L2ROV.
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Eduardo CAMPOS MERCADO received the B.S. degree inelectromechanical engineering from the ITZ (Instituto Tec-
nológico de Zacatepec) in 2008, and the M.S. degree inautomatic control from the CINVESTAV (Centro de Investi-
gación y de Estudios Avanzados del IPN), México, in 2010.He received his Ph.D. degree in 2014 from CINVESTAV and
LIRMM (Laboratoire d’Informatique, de Robotique et de
Microélectronique de Montpellier). Currently he is work-ing in the development of the AUV (Autonomous Under-
water Vehicle) and artificial vision application in the un-derwater robot.
Ahmed CHEMORI received his M.Sc. and Ph.D. degrees
respectively in 2001 and 2005, both in automatic con-trol from the Grenoble Institute of Technology. He has
been a Post-doctoral fellow with the Automatic control
laboratory of Grenoble in 2006. He is currently a tenuredresearch scientist in Automatic control and Robotics at
the Montpellier Laboratory of Informatics, Robotics, andMicroelectronics. His research interests include nonlin-
ear, adaptive and predictive control and their applica-tions in humanoid robotics, underactuated systems, par-
allel robots, and underwater vehicles.
Vincent CREUZE received his Ph.D. degree in 2002 inrobotics from the University Montpellier 2, France. He
is currently an associate professor at the University
Montpellier 2, attached to the Robotics Department ofthe LIRMM (Montpellier Laboratory of Computer Sci-
ence, Robotics, and Microelectronics). His research inter-ests include design, modelling, and control of underwater
robots, as well as underwater computer vision.
Jorge A. TORRES MUÑOZ was born in Mexico City, onMay 13, 1960. He received the B.S. degree in Electronic
Engineering from the National Polytechnic Institute (IPN)
of Mexico in 1982, the M.S. degree in Electrical Engineer-ing from CINVESTAV-IPN, Mexico in 1985, and the Ph.D.
degree in Automatic Control from LAG, INPG, France, in1990. He joined the Department of Electrical Engineering
at the CINVESTAV, Mexico, in 1990. He spent a sabbati-cal year, from September 1997 to August 1998, at the In-
stitute of Research in Communications and Cybernetics,IRCCYN-Nantes, France. Then, he served has the head of
the Department of Automatic Control since its creation in
September 1999 until January 2003, when he was calledo serve as Secretary of Planning as a member of the Direction team of CINVESTAV,
ntil March 2004. He was leading, from the Mexican side, the French Mexican Lab-ratory on Applied Automation (LAFMAA) of CNRS from January 2002 to January
006. He was nominated as Deputy Director of the UMI 3175 LAFMIA at CINVES-AV Mexico, which is a joint research laboratory founded by CNRS, CINVESTAV and
ONACYT for the period 2008–2012. His research interest lies in the structural ap-
roach of linear systems, stability of multivariate polynomials, and control of bio-rocess for waste water treatment and control of mini-submarines.
Rogelio LOZANO joined the Department of Electrical En-
gineering at the CINVESTAV, Mexico, in 1981 where he
worked until 1989. He was head of the Section of Auto-matic Control from June 1985 to August 1987. He has held
visiting positions at the University of Newcastle, Australia,from November 1983 to November 1984, NASA Langley
Research Center VA, from August 1987 to August 1988,and LAG, France, from February 1989 to July 1990. He is a
CNRS Research Director since 1990. R. Lozano was Asso-ciate Editor of Automatica from 1987 to 20 0 0 and of Int. J.
of Adaptive Control and Signal Processing since 1993. He
was head of the Laboratory Heudiasyc, UMR 6599 CNRS-UTC from January 1995 to December 2007. Since April
008 he is the head of the UMI 3175 LAFMIA at CINVESTAV Mexico, which is aoint research laboratory founded by CNRS, CINVESTAV and CONACYT.