Optimization techniques for autonomous underwater vehicles: a
practical point of view
M. Chyba, T. Haberkorn
Department of Mathematics
University of Hawaii, Honolulu, HI 96822
Email: [email protected],
[email protected]
Abstract— The main focus of this paper is to design time
efficient trajectories for underwater vehicles. The equations
of motion are derived from Lagrangian mechanics and we
assume bounds on the controls. We develop a numerical
algorithm based on the time optimal trajectories. Our goal
is to design controls that are piecewise constants with a small
number of switchings and such that the duration for the
vehicle to achieve the trajectory differs from the time optimal
one by a number negligible in terms of the application.
I. I NTRODUCTION
In the exploration of our world’s oceans, there comes a
time when human controlled vehicles are not up to the task.
Whether it be deep ocean survey, or long term monitoring
projects, Autonomous Underwater Vehicles (AUV’s) are
the natural alternative. One of the essential components
for an underwater vehicle to be autonomous is the ability
to design its own trajectories. It is known as the motion
planning problem. Moreover, the goal is to find trajectories
that are efficient with respect to some criteria. Indeed, we
may want our vehicle to stay down longer for monitoring,
or finish the search faster to save money. All of these
questions can be addressed through optimal control. Much
research has been allocated to AUV propulsion, robustness,
sensors, and power sources. With the plethora of sizes and
shapes combined with the youth of the AUV field, and the
complexity of hydrodynamics, accurate models of AUV’s
are hard to come by. Thus, among all AUV research areas,
the area of optimal control is one of the least developed.
Almost all existing controllers have the disadvantage that
they do not take into account minimization criteria such
as time and energy. In this paper, we address the problem
of finding the time minimal trajectories for underwater
vehicles; it is an optimal control problem (OCP). Previous
results were obtained in [2], [3], [4], [5], [6], [7] about
the structure of time minimal trajectories and the role of
the singular trajectories. However, due to the complexity of
the equations we need to supplement our theoretical study
with numeric computations. It is the subject of this paper. A
first way to obtain numerically the desired trajectories is to
discretized our OCP and to solve the resulting optimization
problem whose unknowns are the discretized state and
control of the system. However, the obtained solutions are
Research supported in part by NSF grant DMS-030641
usually not feasible in terms of practical implementation
since their control structures have a lot of switchings.
Another drawback of this method is that the resulting
optimization problem is time consuming to solve since it
needs a large number of unknowns in order to have the
desired accuracy. For these reasons, we propose another
approach that consists in looking for control strategies that
are time optimal with respect to a given switching structure.
To accomplish that goal, we fix the number of switching
times and rewrite the OCP as an optimization problem
whose unknowns are the switching times and the value of
the constant arcs for each component of the control. This
method has two main advantages. The first one is that such
optimization problem has a small dimension and that we
obtain a much better accuracy since we can use high order
integration scheme to integrate the dynamical system of
the model. The second, and main, advantage is that the
obtained control strategies are easier to implement than the
ones resulting from the solving of the discretized OCP on a
real model. Indeed, the former ones are just concatenation of
constant arcs for the control and we can even fix a relatively
small number of switching times so that real thrusters can
follow these strategies more efficiently. Furthermore, it is
also possible to add some smoothing at the switching times
so that the junctions between constant thrust arcs become
more regular.
II. M ODEL
In this section, using Lagrangian mechanics, we derive
the equations of motion for marine vehicles in 6 degrees of
freedom in the body-fixed frame. Let us first introduce some
notations. We denote by η = (x, y, z, φ, θ, ψ)t the position
and orientation of the vehicle with respect to the earth-fixed
reference frame, the coordinates φ, θ, ψ being the Euler
angles for the body frame. The coordinates corresponding to translational and rotational velocities in the body
frame are ν = (u, v, w, p, q, r)t . See Figure 1. We define
η1 = (x, y, z)t , η2 = (φ, θ, ψ)t and ν1 = (u, v, w)t , ν2 =
(p, q, r)t . We then have the following relation
η̇ = J(η)ν
(1)
where J represents the linear and angular velocity transformations. Let L = T − V be the Lagrangian, with
and with the following control vector fields:
p
u
r
z
G(χ) =
q
v
w
0
M −1
(10)
In the sequel we will call χ the state of the system while
the variable η represents a configuration of the system.
y
x
Fig. 1.
Constraints on the control
Earth-fixed and body-fixed reference frame.
T = TRB + TA ; respectively the rigid-body kinetic energy
and the fluid kinetic energy. The potential V is defined
implicitly by
∂V
= J −1 (η)g(η)
(2)
∂η
where g(η) represents the restoring forces (gravitational
forces and moments). Let us recall first the quasi-Lagrange
equations (a generalization of Kirchoff’s equations when
considering a non zero potential):
d ∂L
∂L
∂L
(
) + ν2 ×
− J1T (η2 )
= τ1
dt ∂ν1
∂ν1
∂η1
(3)
∂L
∂L
∂L
d ∂L
(
) + ν2 ×
+ ν1 ×
− J2T (η2 )
= τ2 (4)
dt ∂ν2
∂ν2
∂ν1
∂η2
where τ = (τ1 , τ2 ) is the vector of control inputs. Let us
determine more precisely these equations. Since ∂T
∂η = 0,
∂L
we have by construction that −J(η) ∂η = g(η). Moreover,
d ∂L
we have that ∂L
∂ν = M ν, dt ∂ν = M ν̇. Defining the Coriolis
and Centripetal matrix as
∂T
ν2 × ∂ν
1
C(ν)ν =
∂T
∂T
ν2 × ∂ν
+ ν1 × ∂ν
2
1
we can rewrite equations (3) and (4) as
M ν̇ + C(ν)ν + D(ν)ν + g(η) = τ
(5)
where the term D(ν)ν accounts for the damping forces. We
obtained:
ν̇
η̇
= −M −1 (C + D)ν − M −1 g + M −1 τ
= J(η)ν
(6)
(7)
where M is the inertia matrix, C the coriolis and centripetal
matrix, D the damping forces and the variable τ represents
the control. See [8] for instance for more details. We can
rewrite equations (6) and (7) as an affine control system:
χ̇(t) = f (χ(t)) + G(χ(t))u(t), χ(t) ∈ IR12 .
where χ = (η, ν), the drift is
J(η)ν
f (χ) =
−M −1 (C + D)ν − M −1 g
(8)
(9)
The controls represent the thrust, and we assume the
following bounds:
U = {τ ∈ IR6 ; αi ≤ τi ≤ βi , αi < 0 < βi , i = 1, · · · , 6}.
(11)
A control is said to be admissible if it is a measurable
bounded function τ such that τ (t) ∈ U for a.e. t.
Let us now describe the parameters used for the numerical computations. The location of the center of buoyancy
and gravity are respectively: (xB , yB , zB ) = (0, 0, 8.1e−4)
(meters) and (xG , yG , zG ) = (0, 0, 5.64e − 4) (meters). The
inertia matrix, M , takes the following form:








3/2 m
0
0
0
mzG
0
0
3/2 m
0
−mzG
0
0
0
0
0
−mzG
3/2 m
0
0
Ix
0
−Ixy
0
−Ixz
mzG
0
0
−Ixy
Iy
−Iyz
0
0
0
−Ixz
−Iyz
Iz








(12)
where m = 120.9091 kg is the mass of the AUV, and I.
and I.. are inertia factors (Ix = 6.8 kg.m, Iy = 7.2 kg.m,
Iz = 9.1 kg.m, Iyz = Ixz = Iyz = 0). The restoring forces
and moment vector g(η) is taken to be:




g(η) = 



(Wg − B) sin θ
(−Wg + B) cos θ sin φ
(−Wg + B) cos θ cos φ
(zG Wg − zB B) cos θ sin φ
(zG Wg − zB B) sin θ
0








where Wg is the weight of the vehicle, and B = Wg + 38
(Newton) is the buoyancy force on the vehicle. The damping
forces D(ν) are of the form −diag(Xu + Xuu |u|, Yv +
Yvv |v|, Zw + Zww |w|, Kp + Kpp |p|, Mq + Mqq |q|, Nr +
Nrr |r|) where (Xu , Xuu ) (= (0, −150)), (Yv , Yvv ) (=
(0, −150)), and (Zw , Zww ) (= (0, −150)) are the drag coefficients for pure surge, sway, and heave, respectively. And,
(Kp , Kpp ) (= (−50, −100)), (Mq , Mqq ) (= (−50, −100)),
(Nr , Nrr ) (= (−50, −100)) are the drag coefficients for
pure roll, pitch, and yaw, respectively.
Finally, the coriolis matrix C is:

0

0


0


−m zG r

 m − w + Zwd w
m v − Yvd v
−m (zG r)
Zwd w − m w
C3,4
0
C5,4
C6,4
satisfies the maximum principle is called an extremal, and
the vector function λ(.) is called the adjoint vector.
The maximum condition (14), along with the control
domain (11), is equivalent to almost everywhere:
0
0
0
0
0
0
m w − Zwd w
C4,3
−m zG r
C5,3
m − u + Xud u −m 0
m (w) − Zwd w
−m zG r
C3,5
C4,5
0
C6,5
C1,6
m u − Xud u
m0
C4,6
C5,6
0
τi (t) = αi if ϕi (t) < 0 and τi (t) = βi if ϕi (t) > 0 (15)








Where Xud = Yvd = Zwd = −m/2 and Kpd = Mqd =
Nrd = 0 and the coefficients C.,. are:
C1,6
C3,4
C3,5
C4,3
C4,5
C4,6
C5,3
C5,4
C5,6
C6,4
C6,5
=
=
=
=
=
=
=
=
=
=
=
m (zG p − v) + Yvd v
m (v − zG p) − Yvd v
Xud u − m (u + zG q)
m (zG p − v) + Yvd v
Iz r − Iyz q − Ixz p − Nrd r,
Iyz r + Ixy p − Iy q + Mqd q
m (u + zG q) − Xud u
Iyz q + Ixz p − Iz r + Nrd r,
Ixz r − Ixy q + Ix p − Kpd p
Iy q − Iyz r − Ixy p − Mqd q,
Ixy q − Ix p − Ixz r + Kpd p
for i = 1, · · · , 6, where ϕi (.) are called the switching functions. The switching functions are absolutely continuous
functions defined as follows (i = 1, · · · , 6):
ϕi : [0, T ] → IR,
ϕi (t) = λt (t)gi (χ(t))
According to (15), the zeroes of the switching functions
define the structure of the time-optimal control τ (.). If
ϕi (t) is zero on a nontrivial subinterval of [0, T ], the corresponding extremal is called τi -singular. The component
τi is then called singular on the corresponding subinterval.
The maximum principle implies that if ϕi (t) 6= 0 for almost
all t ∈ [0, T ] the component τi of the control is bang-bang,
which means that τi only takes values in {αi , βi } for almost
every t ∈ [0, T ]. A time ts ∈ (0, T ) connecting a singular
and a bang arc, or two bang arcs (with different values) is
called a switching time.
To study the zeroes of the switching functions and the
existence of singular extremals we analyze their derivatives.
It follows from ϕi (t) = λt (t)gi , that almost everywhere in
t the following is verified:
ϕ̇i = −λt [f, gi ](χ) −
The domain of control is assumed to be U = [−20, 20] ×
[−20, 50] × [−5, 5]3 Newton.
Let χ0 , χT be initial and final states. Assume that there
exists an admissible time-optimal control τ : [0, T ] → U
such that the corresponding trajectory χ(.) solution of (8)
steers the system from χ0 to χT .
The maximum principle, [1], implies that there exists an
absolutely continuous vector λ : [0, T ] → IR12 , λ(t) 6= 0
for all t, such that the following conditions hold almost
everywhere:
∂H
χ̇j =
(χ, λ, τ ),
∂λj
∂H
λ̇j = −
(χ, λ, τ )
∂χj
(13)
H(χ, λ, τ ) = λt f (χ) +
6
X
(17)
ϕ̇i (t) = −λt (t)[f, gi ](χ(t)).
(18)
Notice that it is a property shared by the set of controlled
mechanical system of the form kinetic minus potential
energy. Using equations (9) and (10) we have that
J(η)Mi−1
[f, gi ] =
(19)
∂
−1
(C + D)ν)Mi−1
∂ν (−M
where Mi−1 denotes the ith column of the inverse of the
inertia matrix. The second derivative is given by:
n
X
λt [gj , [f, gi ]](χ)τj
(20)
j=1
λt gi (χ)τi
i=1
is the Hamiltonian function where the gi are the column
vectors of G. Furthermore, the maximum condition holds:
H(χ(t), λ(t), τ (t)) = max H(χ(t), λ(t), γ)
γ∈U
λt [gj , gi ](χ)τj .
Since the control vector fields are given by the inverse of
the inertia matrix, their are constant. A consequence is that
ϕ̇i is an absolutely continuous function given by:
ϕ̈i = λt ad2f gi +
for j = 1, · · · , 12, where
n
X
j=1
2
III. M AXIMUM P RINCIPLE
(16)
(14)
Moreover, the maximum of the Hamiltonian is constant along the solutions of (13) and must satisfy
H(χ(t), λ(t), τ (t)) = λ0 , λ0 ≥ 0. A triple (χ, λ, τ ) that
where ad2f gi stands for [f, [f, gi ]]. Notice that since the
first n components of the vector field [f, gi ] do not depend
on ν but only on η, we have that the first n components of
[gj , [f, gi ]] are zeroes. As a consequence, in the case we
make the assumption that the damping matrix is linear in
ν, we have that the vector fields [gj , [f, gi ]] can be written
as a linear combination of the gi .
Unless we make additional assumptions on the matrices
involved in the equations of motion of the vehicle, which
x (m)
2
4
6
8
10
12
4
2
0
0
2
4
6
8
10
12
w (m/s)
z (m)
0.5
0
0
2
4
6
8
10
12
1
p (rad/s)
φ (rad)
−0.5
0
−1
0
2
4
6
8
10
12
q (rad/s)
0.5
0
−0.5
0
2
4
6
8
10
12
0
r (rad/s)
θ (rad)
0
v (m/s)
y (m)
0
METHOD
−0.5
−1
0
2
4
6
t (s)
Fig. 2.
8
10
12
0.5
0
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
t (s)
8
10
12
1
0.5
0
0.5
0
−0.5
0.1
0
−0.1
0.1
0
−0.1
0.1
0
−0.1
Time optimal trajectory for (N LP )
1
20
τ
0
−20
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
t (s)
8
10
12
2
20
τ
0
−20
3
50
τ
0
−50
τ
4
5
0
−5
τ
5
5
0
−5
5
6
They are two kind of methods to solve numerically an
optimal control problem: the indirect and the direct. Due to
the form of our problem, the use of indirect methods is very
complicated without knowing the structure of the solution
in advance. We then use a direct method. It consists in
discretizing (OCP ) in order to transform it into a nonlinear
optimization problem. We call this new problem (N LP ).
The unknowns of (N LP ) are the discretized states and controls. Its nonlinear constraints come from the discretization
according to a fixed step integration scheme, here a Heun
one, of the dynamic (8). Additional constraints are the ones
on the final state and the upper and lower bounds on the
controls.
To solve (N LP ), we write it using the AMPL modeling
language, see [9], and we apply the large-scale nonlinear
solver IpOpt, see [13]. The solving usually takes around
30 min, counting refining steps, for a time grid of 4000
points (so approximatively 72000 unknowns and 48000
nonlinear constraints). Notice that a solution of (N LP )
is only an approximation of a solution of (OCP ) since
the fixed step integration scheme used to discretized the
dynamic introduces inaccuracy but mostly because the
control is piecewise constant due to the discretization of
the problem. To increase the accuracy, we need to refine the
time discretization grid which also increases the number of
unknowns of (N LP ). A direct effect of a larger number
u (m/s)
5
τ
IV. O PTIMIZATION
of unknowns is an increase in the execution time needed to
solve (N LP ), and ultimately a failure to solve it without
important computational resources.
Figure 2 shows an example of a solution of (N LP ) for
χ0 = (0, 0, · · · , 0) and χT = (5, 3, 0, · · · , 0) for the model
described previously. The evolution of the corresponding
controls are represented on Figure 3. The time for this
ψ (rad)
would result in an unrealistic model, the complexity of
the equations is such that an analytical study is very
difficult. Partial results can be derived but we cannot expect
general results such as a uniform bounds on the number of
switchings. Moreover, our major goal is the implementation
of our techniques to an existing vessel. We then have to
take into account practical constraints such as the capacity
of the thrusters to follow a prescribed control and the ability
of the vehicle to realize a given path. It is a well known
fact that optimal trajectories can involve singular trajectories or have infinitively many switchings. In our problem,
the complexity of the optimal structure has been outlined
already on a simplified model where we constructed a
semi-camonical form for our system in a neigborhood of
singular trajectories and deduce the existence of chattering
extremals, see [2]. It is a consequence of the fact that a
τi -singular extremal, with i = 4, 5 or 6 is of order 2.
d2q
It means that the lowest order derivative dt
2q ϕi in which
τi appears explicitely with a nonzero coefficient is given
by 2q = 4. Hence, to compute a singular control τi , we
must differentiate the corresponding switching function ϕi
4 times and solve this final equation for the control. Such
a control τi is then computed as a feedback of the state
χ, of the adjoint vector λ and of the other component of
the control (assuming they are constants). For this reason,
we developed a numerical methods taking into account the
criterion to minimize as well as the ability of the vehicle
and the thrusters to follow a prescribed path.
0
−5
Fig. 3.
Time optimal control strategy for (N LP )
trajectory is tfmin ≈ 12.441 s. Looking at the Figure
3, it is obvious that the control strategy is inappropriate
to a practical implementation on real vehicle for at least
two reasons. The first one comes from the existence of
singular arcs for the component τ6 . Along these singular
arcs, the control varies continuously which is impossible to
implement. The second reason is that there is a large number
of switching times. In practice we observe delays with the
thrusters to respond to a given command, as a consequence
we want to keep the number of switchings small. Moreover,
the fact that the trajectory is computed through a fixed step
integration make it less accurate to the theoretical model
than a proper high-order, adaptative step integration.
R EMARK 4.1: The reason to use a high order scheme
to improve the accuracy of the solution of the (N LP )
problem is that we already have a poor knowledge of the
hydrodynamics coefficients which introduces uncertainties
in the experiments, hence we would like to minimize any
other source of potential difference between the theoretical
computations and the experiments.
Inspired from the work in [10], [12], we develop another
approach to overcome the main issues of the previous solving method. In [12], the authors use the discretized solution
of an optimal control problem to extract the switching
structure of the optimal control. The next step is to rewrite
the optimal control problem as a nonlinear optimization
problem whose unknowns are the switching times (or more
precisely the time length between two switching times).
They are then able to integrate their dynamical system with
a high order integrator. The motivation for their approach
is the verification of second order sufficient conditions for
optimality. This method cannot be applied to our problem.
The reason is that along time minimal trajectories we
have the existence of singular arcs. As mentioned before,
along these arcs the singular control can be computed as a
feedback of the state and of the adjoint vector, contrary to
[12] where the singular control is a feedback of the state
only. Moreover, the switching structure of the solutions of
(N LP ) is not always easy to extract, see Figure 3. Our
motivation is different, we want to develop an algorithm
to compute trajectories that are efficient in time but mostly
feasible by the vehicle. Hence, our goal is, based on our
previous computations for the (N LP ) problem, compute
trajectories that are close in time from the optimal one
but that are such that their implementation on a vessel
is possible. To reach our goal, we rewrite (OCP ) as an
optimization problem but without taking into account the
switching structure given by the solution of (N LP ). We
fix the number of switching times, preferably to a small
number. The new optimization problem, called (ST P P )p
(for Switching Time Parameterization Problem), has then
for unknowns the time arclengths between two switching
times (plus the time arclength between the last switch and
the final time) but also the values of the constant thrust arcs.
(ST P P )p has the

min tp+1


z∈D



t0
=




 ti+1 =
χi+1 =



χp+1 =




z
=



D
=
following form:
0
ti + ξi , i = 1, · · · , p
Rt
χi + tii+1 χ̇(t, τ i )dt
χT
(ξ1 , · · · , ξp+1 , τ 1 , · · · , τ p+1 )
n(p+1)
IR+
× U p+1
(21)
where ξi , i = 1, · · · , p + 1 are the time arclengths and
τ i ∈ U, i = 1, · · · , p + 1 are the values of the constant
thrust arcs. χ̇(t, τ i ) is the right hand side of the dynamical
system (8) with the constant control τ i .
The integration of the dynamical system of (ST P P )p
(which is simply (8)) can be done with a high order adaptative step integrator. We use DOP853, see [11]. (ST P P )p
has a much smaller number of unknowns than (N LP ) and
so, even if the integration takes more time because it is
more accurate, the computational resources needed to solve
(ST P P )p are drastically reduced. To solve it we use once
more IpOpt, see [13], which yields very good results.
V. N UMERICAL
RESULTS
In this section we describe the results using the method
described in the previous section. One of the main advantage of (ST P P )p is that we are usually able to fix a small
p, that is a small number of switching times. However,
notice that a solution of (ST P P )p will only be optimal
with respect to the fixed number of switching times. So
it won’t be a solution of (OCP ), but only an admissible
control and trajectory that steers the system from χ0 to χT .
Figure 5 shows a control strategy solution of (ST P P )3
with the same terminal configurations as for Figure 2, i.e.
χ0 = (0, 0, · · · , 0) and χT = (5, 3, 0, · · · , 0). We impose
the total number of switching times along the trajectory to
be 3. The final time corresponding to the solution shown
on Figure 5 is tf ≈ 12.814 s. This final time has to be
compared to the one of a solution of (N LP ), Figure 2,
which is tfmin ≈ 12.441 s. We see that the difference
is neglectable providing that the obtained longer control
strategy is easier than the time-optimal one. A solution
of (ST P P )1 yields a tf ≈ 17.445 s and a solution of
(ST P P )2 yields tf ≈ 13.028 s.
Note that along the solution of (ST P P )3 the control
strategy is not bang-bang. Indeed, since the values of
the control between two switching times belong to the
unknowns of the problem, there is no guarantee that the
thurst of the vehicle will be saturating all the time, i.e. take
their values in {αi , βi }.
VI. C ONCLUSION
In this paper, we focus on a practical implementation of
time optimal trajectories for an underwater vehicle. Time
optimal trajectories are often difficult to implement due to
their structure, moreover in practice we are not interested
x (m)
u (m/s)
5
0
0
5
10
v (m/s)
y (m)
4
2
0
0
5
10
0.5
5
10
φ (rad/s)
0
−0.5
0
5
10
θ (rad/s)
0.4
0.2
0
0
5
10
0
ψ (m/s)
ψ (rad)
0
0.5
θ (rad)
φ (rad)
0
−0.5
−1
0
5
10
0
0
5
10
0
5
10
0
5
10
0
5
10
0
5
10
0.5
0
−0.5
w (m/s)
z (m)
1
0.5
0.4
0.2
0
0.1
0
−0.1
0.1
0
−0.1
0.1
R EFERENCES
0
−0.1
0
5
t (s)
Fig. 4.
10
t (s)
Trajectory corresponding to (ST P P )3 .
1
20
τ
0
−20
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
0
2
4
6
8
10
12
2
20
τ
0
−20
3
50
τ
0
−50
τ
4
5
0
−5
τ
5
5
0
−5
τ
6
5
0
−5
t (s)
Fig. 5.
Computations on the model presented in this paper have
showned that usually no more than 5 switching times are
necessary to obtain a solution with a final time very close
to the optimal one. We are currently testing our algorithms
on an underwater vehicle, the results will be presented in
a forthcoming article. Notice that some additional issues
may have to be taken into account. Indeed, underwater
propeller, like any other engines, are not able to accomplish
instantaneous switchings. However, since the structure of
(ST P P )p allows flexibility to additional constraints, we
can also adapt the switching time parameterization in order
to impose continuous junctions between the constant arcs
for the control. For instance, one can introduce a linear
junction.
Control strategy corresponding to (ST P P )3 .
in the time optimal trajectory but in a trajectory efficient in
time and easily feasible by the vehicle. Our method allows
us to restrict the control strategy to piecewise constant
functions with a small total number of switching times. The
essential idea for the algorithm to work was to introduce
the value of constant arcs for the control has unknowns
in our optimization problem. The results exceeded our
expectations.
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