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Input-Constrained Path Following for Autonomous Marine Vehicles with a Global Region of Attraction

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  • Instituto Superior Tecnico, University of Lisbon

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This paper presents a solution to the problem of path following control for autonomous marine vehicles (AMVs) subject to input constraints and constant ocean current disturbances. We propose two nonlinear control strategies: the first is obtained by using a Lyapunov-based design method, while the second is developed by adopting a Model Predictive Control (MPC) framework. We show that, with the proposed control strategies, the path-following error is globally asymptotically stable (GAS). Simulations with a kinematic model of the vehicle support the theoretical results. Simulations with a realistic model of the Medusa class of AMVs show the robustness of the proposed control strategies.
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Input-Constrained Path Following for
Autonomous Marine Vehicles with a Global
Region of Attraction ?
Nguyen T. Hung F. Rego ,∗∗ N. Crasta A. M. Pascoal
Laboratory of Robotics and Systems in Engineering and
Science(LARSyS), ISR/IST, University of Lisbon, Lisbon, Portugal
{hungnguyen,frego,ncrasta,antonio}@isr.ist.utl.pt.
∗∗ Automatic Control Laboratory, EPFL, Switzerland
Abstract: This paper presents a solution to the problem of path following control for
autonomous marine vehicles (AMVs) subject to input constraints and constant ocean current
disturbances. We propose two nonlinear control strategies: the first is obtained by using a
Lyapunov-based design method, while the second is developed by adopting a Model Predictive
Control (MPC) framework. We show that, with the proposed control strategies, the path-
following error is globally asymptotically stable (GAS). Simulations with a kinematic model
of the vehicle support the theoretical results. Simulations with a realistic model of the Medusa
class of AMVs show the robustness of the proposed control strategies.
Keywords: AMVs, Path following, MPC, Input constraint.
1. INTRODUCTION
Along with point stabilization and trajectory tracking,
path following is one of the fundamental motion control
problems of autonomous marine vehicles, which are nor-
mally under-actuated, see (De Luca et al. (1998)) for an
overview of these problems. Path following is useful in a
number of practical applications that include, for example,
mapping of underwater habitats, where the key objective
is to obtain a map of a given area without overly restrictive
temporal specifications. From a technical standpoint, when
compared to trajectory tracking, path following has the
potential to exhibit smoother convergence properties and
reduced actuator activity (Aguiar and Hespanha (2007);
De Luca et al. (1998)).
The last two decades have witnessed a considerable inter-
est in the path following problem because of its wide range
of practical applications and theoretical challenges. By
formulating the path following problem as a stabilization
problem in a conveniently defined path following error
space, a large number of methods have been proposed to
asymptotically stabilize the path following error system
about the origin; see for example (Caharija et al. (2016);
Lapierre et al. (2006)) and the references therein. In this
context, Lyapunov-based techniques - that target explic-
itly the design of stabilizing control laws for nonlinear
systems - have become a popular tool to solve several
variants of the path following problem. However, many
of the results published on the use of Lyapunov-based
design techniques do not take input constraints directly
into account. As a consequence, proper care must be taken
to ensure that the resulting systems end up operating in a
?This research was supported in part by the Marine UAS project under the
Marie Curie Sklodowska grant agreement No 642153, the H2020 EU Marine
Robotics Research Infrastructure Network (Project ID 731103), and the FCT
Project UID/EEA/5009/2013.
small region where the control law for the unconstrained
system does not violate the constraints.
In recent years, thanks to its ability to handle input and
even state constrains explicitly, Model Predictive Control
has become the method par excellence to solve more chal-
lenging versions of the path following problem, see for
instance (Alessandretti et al. (2013); Yu et al. (2015)). In
this set-up, a stabilizing MPC for a path following error
system is designed with a terminal cost and a terminal
set so as to guarantee recursive feasibility and stability
of the MPC scheme. In (Yu et al. (2015)), the terminal
set is designed locally using a polytopic linear differential
inclusion method. The approach reported in (Alessandretti
et al. (2013)) aims to enlarge the terminal set by resorting
to the stabilizing nonlinear path following control law
proposed in (Aguiar and Hespanha (2007)). The main
drawback of these two approaches is the presence of the
terminal set, which implies that the region of attraction
is local. Furthermore, it is very difficult to characterize
the region of attraction to determine the region where the
vehicle should start.
Motivated by these considerations, in this paper we pro-
pose two control strategies for the path following problem
of AMVs subject to input constraints and disturbances
caused by constant ocean currents. The main contribution
of the paper lies in the fact that the proposed control
strategies are not only able to handle the input constraints
of the vehicle explicitly but also yield a global region of
attraction, meaning that regardless of any initial position
and orientation, it is always guaranteed the vehicle con-
verges to the path asymptotically. In addition, in contrast
with the assumptions in (Alessandretti et al. (2013)), the
vehicle’s speed is constrained to be non-negative. This is
extremely important not only for certain types of marine
vehicles but also for fixed wing UAVs. Finally, constant
ocean currents are also taken explicitly into account.
The paper is organized as follow. Section 2 presents the for-
mulation of the input-constrained path following problem.
Section 3 describes the design and theoretical results of the
proposed path following control strategies. Simulations are
presented in Section 4. Finally, Section 5 contains the main
conclusions.
2. INPUT-CONSTRAINED PATH FOLLOWING
In what follows, {I} ={xI, yI, zI}and {B} ={xB, yB, zB}
denote an inertial North-East-Down (NED) frame and a
body-fixed frame, respectively. In the marine literature, see
Fossen (2011), the axis xInormally points to the North,
the axis yIpoints to the East, and the axis zIpoints down-
ward, normal to the Earth’s surface, . Let Pbe the center
of mass of the vehicle and denote by p= [x, y, z]TR3
the position of Pin {I}. For simplicity of exposition,
we assume that the vehicle moves in a horizontal plane;
without loss of generality, we make z= 0. Let Pbe a
planar path that the vehicle must follow, parameterized
by its arc length s.
We assume the motion of the vehicle is disturbed by
a constant ocean current represented by a vector vc=
[vcx, vcy ]TR2expressed in the inertial frame {I}. The
3-DOF kinematic model of the vehicle is described by
˙x=urcos ψvrsin ψ+vcx,
˙y=ursin ψ+vrcos ψ+vcy,
˙
ψ=r,
(1)
where ur, vrdenote the surge and sway speed components,
respectively, of the velocity vector vb= [ur, vr]Tof the
vehicle with respect to the fluid, expressed in the body
frame, ψis the yaw angle, and ris the yaw rate. For
a large class of under-actuated AMVs such as Medusa
and Delfim(Abreu et al. (2016), or Charlie (Bibuli et al.
(2009)) the sway speed is in practice so small that it can be
neglected. For this reason, in this work we consider vr= 0
and therefore vb= [ur,0]T. We denote by v= [ ˙x, ˙y]Tthe
total velocity vector of the vehicle in the inertial frame
{I} and its magnitude (inertial speed of the vehicle) by
U:= kvk. We also denote by χthe course angle, defined
by the orientation of the total velocity vector in the inertial
frame, defined as χ= arctan 2( ˙y, ˙x). With this notation,
the kinematics of the AMV can be rewritten as
˙x=Ucos(χ),˙y=Usin(χ) (2a)
˙χ=rχ=r1kvckcos(αψ)
m˙
Ukvcksin(αψ)
Um ,
(2b)
where rχdenotes the course rate, α= arctan 2(vcy , vcx),
and m=qU2− kvck2sin2(αψ) (see the Appendix for
details). Notice that, without ocean current, (2) resembles
those of a unicycle model (Lapierre et al. (2006)) with
input vector (ur, r). Notice also that when kvck= 0, that
is, in the absence of ocean currents, rχ=r. We assume that
the total speed Uand the course rate rχcan be tracked
by an autopilot that admits input commands in the set
Uv:= {(U, rχ) : 0 UUmax,|rχ| ≤ rχmax },(3)
where Umax, rχmax are saturation values. Notice that, in
practice, it is common that the autopilot admits the
heading rate rather than the course rate. Given rmax (a
constraint on the heading rate such that |r| ≤ rmax),
kvckmax (an upper bound on the magnitude of the ocean
current), and Umin (a lower bound of the total inertial
speed of the vehicle), rχmax can be easily computed from
(2b) such that for any angle (αψ), if the computed
course rate lies in the constraint set, then the heading
rate computed from the course rate and (2b) satisfies the
heading rate constraint. Obviously, without ocean current
the course angle coincides with the heading, and therefore
rχmax =rmax.
Referring to the Fig. 2.1, let Sbe an arbitrary point on
the path and let sbe the signed curvilinear abscissa of S
along the path which, we recall that, is parameterized by
its arc length. Attached to S, consider a parallel transport
frame {F} ={xF, yF, zF}with the origin at Sand its axis
defined as follows: xFis aligned with the tangent to the
path at Sand points in the direction of increasing path
length, zFpoints downward to the Earth’s surface, and
yFis determined by the positive right-hand rule. In the
set-up adopted for path-following, the parallel transport
frame moves along the path in a manner to be determined
and plays the role of a virtual “reference” for position and
course angle that the vehicle must track to achieve good
path following. This yields an extra degree of freedom
that allows us to use the linear speed ˙sof the virtual
“reference” along the path as a control input to aid in
the solution of the path following problem. See (Lapierre
et al. (2006)) and the references therein for an introduction
to this methodology.
As illustrated in Fig. 2.1, given an arbitrary point Son
yF
ψF
xI
yI
xF
p
pd
eF
S
ey
ex
yB
ψ
v
xB
P
χ
vb
vc
Fig. 2.1. Vehicle position expressed in the reference frame.
vcis the ocean current. The sway velocity is neglected,
i.e. vr= 0.
the path we let eF= [ex, ey,0]Tbe the position vector
of the vehicle expressed in the parallel transport frame at
S, that is, the position error between the vehicle and the
path. Let κ(s) and ψFbe the curvature of the path and
the angle that the tangent to the path makes with xI,
respectively, at S. Clearly, the orientation error between
the vehicle and the parallel transport frame is given by
ψe=χψF. Let v= ˙sbe the speed of the virtual
“reference”, x= [ex, ey, ψe]Tthe path following error
vector containing the position and orientation errors and
u= [U, v, rχ]Tthe input vector. Using the methodology
exposed in Lapierre et al. (2006) for wheeled robots,
straightforward computations show that the evolution of
the path following error vector is described by the dynamic
equation
˙
x=f(x,u) = "v(1 κ(s)ey) + Ucos(ψe)
κ(s)exv+Usin(ψe)
rχκ(s)v#.(4)
Later, as we shall see, for design purposes, we impose
a constraint on v, that is, |v| ≤ vmax, where vmax is a
parameter that will be specified later.
We are now in a position to formulate the following input-
constrained path following problem.
The Input-Constrained Path Following Problem.
Given a spatial path Pparameterized by its arc length s, a
positive desired speed profile vd(s)along the path, and the
input constraint set Uvdefined by (3), derive a feedback
control law for (U, rχ)Uvand v, with |v| ≤ vmax so as
to drive the path following error dynamics described in (4)
asymptotically to zero, while ensuring also asymptotically
that vtracks the desired speed profile vd(s), that is, v(t)
vd(s(t)) tends to zero.
3. MAIN RESULTS
In this section, two input-constrained nonlinear controllers
are proposed to solve the constrained path following prob-
lem. The first controller is developed by resorting to a
Lyapunov-based method, whereas the second is derived
using MPC.
3.1 Input-Constrained Path Following with a Lyapunov
Based Controller
In what follows we require that the input constraints,
the path specifications, and the ocean current satisfy the
following assumptions.
Assumption 1.
A1.1 The vehicle inner loops guarantee that the references
for Uand r(and therefore rχ) can be tracked
accurately. Thus, we neglect the dynamics of the
inner loops.
A1.2 The desired speed profile is assigned so that
max(vd(s)) Umax and min(vd(s)) >kvckmax.
A1.3 The constraint imposed for the speed of the virtual
reference, vmax, satisfies vmax >max(vd(s)).
A1.4 The curvature of the path is sufficiently smooth and
bounded such that the angular rate of the path
satisfies max(κ(s))vmax < rχmax.
A1.5 The ocean current can be either measured or esti-
mated.
Remark 1. Assumption A1.2 implies that if the current
is strong, the desired speed profile should be large enough
so as to make the vehicle move forward. In addition, it
also implies that the surge speed uris always positive, thus
forcing the vehicle to always move forward with respect
to the water. Assumptions A1.3 and A1.4 are included
to ensure that the vehicle and the virtual reference can
“catch up with each other”. Assumption A1.5 allows us to
compute the heading rate rfrom rχ. In practice, because it
is difficult to measure the ocean current, one can estimate
it using the kinematic model of the vehicle given by (1),
see the current estimator in (Vanni et al. (2008)).
We next show that there exists a global nonlinear control
law that satisfies the input constraints and stabilizes the
path following error dynamics described by (4).
Theorem 1. (Lyapunov-based controller)
Consider the path following error system (4) subjected to
the vehicle input constraint given by (3) and |v| ≤ vmax .
Further, let Assumption 1 hold true. Then, the Lyapunov-
based control law given by
uL(x) = "U
v
rχ#=
vd(s)
Ucos(ψe) + k1tanh(ex)
keyUsin(ψe)
(1+e2
x+e2
y)ψek2tanh(ψe) + κ(s)v
(5)
solves the input constrained-path following problem, where
k1, k2, k R>0are tuning parameters that satisfy
0< k1vmax max(vd(s)),
0.5kmax(vd(s)) + k2rχmax max(κ(s))vmax.(6)
Furthermore, the control law given by (5) renders the
origin of (4) globally asymptotically stable.
PROOF: Feasibility. Since vd(s)>0 and Assumption
A1.2 holds, Utrivially satisfies (3). From (5), it follows
immediately that
|v|=|vd(s) cos ψe+k1tanh(ex)| ≤ vd(s) + k1.
Now Assumption A1.3 implies that (vmax max(vd(s)) >
0. Let k1>0 be such that (6) satisfied. Then, clearly
|v| ≤ vmax. As for the constraint on the course rate, we
note the inequality |ey/(1 + e2
x+e2
y)| ≤ 1/2 and the fact
that sin(ψe)e1 holds for all ex, ey, ψe. Using these
facts, we have
|rχ|=
keyUsin ψe
(1 + e2
x+e2
y)ψe
k2tanh(ψe) + κ(s)v
0.5kmax(vd(s)) + k2+ max(κ(s))vmax.
Thus, by proper selection of the positive constants k, k2
satisfying (6), we can conclude that rχrχmax.
Global Asymptotic Stability. We consider the Lyapunov
function candidate given by
VL(x) = k
2log(1 + e2
x+e2
y) + 1
2ψ2
e(7)
with k > 0 as given above. Computing the derivative of the
Lyapunov function and replacing the inputs by the control
law (5) yields
˙
VL=kk1extanh(ex)
1 + e2
x+e2
y
k2ψetanh(ψe)0,x.
Observe that ˙
VL= 0 when ex= 0 and ψe= 0. We now
consider the set Ω := {xR3|ex= 0, ψe= 0}. In Ω,
since lim
ψe0sin ψee= 1, replacing the input uin (4) by
(5), we obtain ˙
ψe=keyU/(1 + e2
y). It is clear that, in
Ω, if ψe(t)0 then it follows that ˙
ψe(t)0, and as
a consequence ey(t)0. Further, since VLis radially
unbounded, invoking corollary 4.2 of LaSalle’s theorem
(Khalil (2002)), we conclude that the origin of the path
following error system is GAS.
3.2 Input-Constrained Path Following with MPC
The Lyapunov-based controller in the previous subsection
solves the input-constrained path following problem. How-
ever, the convergence of the path following error to origin
can be slow (as can be seen from the derivative of the Lya-
punov function in the proof of Theorem 1, the convergence
rate is slow when ex, ψeare small, regardless of ey). In
this subsection, we adopt a sampled-data MPC framework
which is also referred as Lyapunov-based MPC (see de la
Pena and Christofides (2008)) to solve the constrained
path following problem and speed up the convergence rate.
To this end, we employ the Lyapunov function (7) and the
Lyapunov-based controller (5) to construct a contractive
constraint that forces the decrease rate of the Lyapunov
function with the MPC controller to be faster than with
the Lyapunov-based controller. In addition, by inheriting
the feasibility and stability with the control law (5), it
follows that terminal constraint and terminal cost, two
ingredients in the general stabilizing MPC framework (see
Mayne et al. (2000)), are no longer necessary.
First, we assign the speed of the vehicle as the desired
speed profile, that is, U=vd(s). The dynamics of the
path following error system (4) can be rewritten as
˙
x=f(x,u) = "v(1 κ(s)ey) + vd(s) cos(ψe)
κ(s)exv+vd(s) sin(ψe)
rχκ(s)v#,(8)
where u= [v, rχ]Tis the input vector that is constrained
in a set Ugiven by
U:= {(v, rχ) : |v| ≤ vmax,|rχ| ≤ rχmax}.(9)
We define a finite horizon open loop optimal control
problem (FOCP), denoted OCP (t, x(t), s(t), Tp), where Tp
is the prediction horizon that the sampled data MPC
solves at every sampling time as follows:
Definition 1. OCP (t, x(t), s(t), Tp)
min
¯
u(·)JTp(x(t), s(t),¯
u(·)) (10)
with
JTp(x(t), s(t),¯
u(·)) := Zt+Tp
t
l(¯
x(τ),¯s(τ),¯
u(τ)) dτ
subject to
˙
¯
x(τ) = f(¯
x(τ),¯
u(τ)), τ [t, t +Tp],¯
x(t) = x(t),(11a)
˙
¯s(τ) = v(τ), τ [t, t +Tp],¯s(t) = s(t),(11b)
¯
u(τ)U, τ [t, t +Tp],(11c)
∂V
xf(x(t),¯
u(t)) ∂V
xf(x(t),un(x(t))) .(11d)
In (11), we use the bar notation to denote the predicted
variables and to differentiate from the actual variables
which do not have a bar. Specifically, ¯
x(τ) is the predicted
trajectory of the path following error and ¯s(τ) is the pre-
dicted value of the path parameter s. Both are computed
using the dynamic model (8) and the initial conditions
(11a) and (11b), driven by the input ¯
u(τ) with τ[t, t +
Tp] over the prediction horizon Tp. We stress that in the
MPC scheme above, we do not require a terminal set and
a terminal cost, which are normally designed to ensure
recursive feasibility and stability. Instead, we add the con-
straint (11d) which is referred as a contractive constraint
to guarantee stability. This constraint is constructed based
on a global Lyapunov function V:R3R0and its as-
sociated stabilizing constrained control law un:R3U.
Finally, l:R3×U×RR0is the stage cost of the
objective functional JTp.
In the sampled data MPC scheme, the optimal control
problem OCP(t, x(t), s(t), Tp) is repeatedly solved at every
discrete sampling instant ti=,iN+, where δis a sam-
pling interval. Let ¯
u(τ), τ [t, t+Tp],be the optimal solu-
tion of the optimal control problem OCP (t, x(t), s(t), Tp).
Then, the MPC control law umpc(·) is then defined as
umpc(t) = ¯
u(t), t [ti, ti+δ].(12)
Before proceeding to the main result for the path following
problem with the proposed MPC scheme, we make the
following assumptions.
Assumption 2.
A2.1 The stage cost l(·) is continuous and l(·) = 0 when
x=0and ua= [v+vd(s) cos ψe, rχκ(s)v]T=0.
A2.2 Given the path following error dynamics in (8), there
exists a global Lyapunov function V:R3R0
such that Vis positive definite and V(x) = 0 if
and only if x=0and an associated nonlinear
feedback control law un:R3Uthat satisfies
∂V
xf(x,un(x)) 0 for all x6=0. Furthermore, un(x)
globally stabilizes (8).
Remark 2. Assumption A2.1 is motivated by the fact that
ua0once the path following error is stabilized.
We are in the position to state the main result of this
subsection.
Theorem 2. MPC for Path Following (MPC-PF).
Consider the path following error system (8) and let As-
sumption 2 holds true. Then, with a sufficiently small sam-
pling interval δ, the proposed sampled data MPC scheme
solves the constrained path following problem. Further-
more, the origin of the resulting path following error sys-
tem is GAS.
PROOF: Recursive feasibility. Since l(·) is continuous
and defined over the compact set U, the optimal control
problem OCP(t, x(t), s(t), Tp) is always feasible. Clearly,
¯
u(τ) = un(x(t)), τ [t, t+δ] is one of the feasible solutions
for ¯
u(τ) for τ[t, t +δ] satisfying constraints (11c) and
(11d), while the rest of ¯
u(τ), τ [t+δ, t +Tp] can be
chosen freely in the input space Usince there are no other
constraints imposed on it.
Global asymptotic stability. Consider the Lyapunov func-
tion Vthat satisfies Assumption A2.2 for the path fol-
lowing error system (8) controlled by the proposed MPC
scheme. The contractive constraint (11d) and Assumption
A2.2 imply that
˙
V(t) = V
xf(x(t),umpc(t)) V
xf(x(t),un(x(t))) 0.
We consider two cases of umpc(t). In the first
case, umpc(t)6=un(x(t)) that results in ˙
V(t) =
∂V
xf(x(t),umpc(t)) < V
xf(x(t),un(x(t))) 0. Obviously,
in this case, Vis strictly decreasing. In the second case,
the MPC scheme finds umpc(t) = un(x(t)). Since un(x)
globally stabilizes (8), x(t) converges to zero as t→ ∞.
We conclude that umpc (t) globally stabilizes (8).
The design of the proposed MPC controller is as follows.
The stage cost is chosen as the quadratic form
l(¯
x(τ),¯s(τ),¯
u(τ)) = xT(τ)Qx(τ) + uT
a(τ)Rua(τ).(13)
In (13), QR3×3denotes a diagonal positive definite
matrix that penalizes the path following error, whereas
RR2×2is a diagonal, positive definite weighting matrix
that penalizes the inputs. This choice of the stage cost
trivially satisfies Assumption A2.1. In what concerns the
existence of a global Lyapunov function and its associated
constrained control law, it is sufficient to verify that the
Lyapunov function VLgiven by (7) and the control law for
(v, rχ) in (5) satisfy the Assumption A2.2.
4. SIMULATIONS
In this section, we present the results of simulations to
show the efficacy of the proposed path following controllers
in solving the input constrained path following problem.
In the first subsection, we use the path following error
system model (4) to verify the main theoretical results
presented in Section 3. In the subsequent subsection, to
show the robustness of the proposed controllers and their
applicability in practice, we perform the simulations with a
realistic model of Medusa, a prototype AMV that has been
intensively used in several European research projects, see
(Abreu et al. (2016)).
4.1 Simulation with a Kinematic Model
We consider the vehicle following the path parameter-
ized by the arc length swith varying curvature κ(s) =
0.1 sin(0.1s)m1. The vehicle inputs are constrained with
0U1.2m s1and |r| ≤ 0.5rad s1. The speed profile
along a path is specified as vd(s) = 0.8+0.2 cos(0.1s)m s1.
Thus, we obtain max(vd(s)) = 1ms1, max(κ(s)) =
0.1m1, rχmax = 0.5rad s1. We set vmax = 1.5m s1,
thus satisfying assumptions A1.3 and A1.4. This implies
feasibility of the path following problem. The tuning pa-
rameters for the nonlinear Lyapunov-based controller (5)
are set to k1= 0.5, k2= 0.175,and k= 0.35. Clearly, they
satisfy (6).
In the proposed MPC scheme, we construct the contractive
constraint (11d) using the Lyapunov function (7) and
the auxiliary stabilizing control law (5) with the above
tuning parameters for the Lyapunov-based controller. The
weighting matrices in the cost function (13) are chosen
as Q=diag(1,1,1) and R=diag(1,10). The sampling time
is chosen as δ= 0.2s and the prediction horizon is set
to Tp= 1s. To solve the finite optimal control problem
OCP(t, x(t), s(t), Tp), we use the open source optimization
framework Casadi and the interior-point solver IPOPT
(Andersson (2013)).
The path following error system is initialized at
[x(0); s(0)] = [20m; 10m; π/2; 0m].The performance of the
proposed controllers is shown in Fig. 4.1. It is visible that
the Lyapunov function is monotonically decreasing over
time and converges to zero with both controllers. However,
due to the contractive constraint, the path following error
controlled by the MPC scheme has faster convergence rate.
The figure also shows that the speed, the heading rate
and the speed of the virtual reference satisfy the input
constraints (black dash lines) with both controllers.
4.2 Simulation with a Realistic Model of the Medusa AMV
To show the applicability of the control method derived
we perform simulations with a realistic model of a Medusa
vehicles. The Medusa model adopted has been intensively
used in simulation work as the last step before the execu-
tion of real time tests with the actual Medusa vehicles at
sea (see (Abreu et al. (2016)) for the details of the vehicle).
We notice that realistic Medusa model contains the kine-
matics model (1) coupled with two autopilots that track
0 50 100 150
Time[s]
0
0.5
1
1.5
2
2.5 Lyapunov function V
0 50 100 150
Time[s]
-0.5
0
0.5
Heading rate r
0 50 100 150
Time[s]
-2
0
2Speed of the virtual reference v
0 50 100 150
Time[s]
0
0.5
1
Total speed of the vehicle U
Fig. 4.1. Performance of the Lyapunov-based controller
(red) and the MPC (blue) scheme for the input
constrained path following problem. Black dash lines
represent the bounds on the inputs.
the desired total speed and heading rate issued by a path
following controller, the dynamics of the Medusa vehicle,
and the dynamics of the thrusters. In the Medusa model,
it is assumed that the total speed Ucan be measured or
estimated using GPS when the vehicle is at the surface, or
using a Doppler in bottom-lock mode when the vehicle is
underwater.
The autopilots admit the reference commands for linear
speed from 0 to 1m s1, and the references for heading rate
from -0.3rad s1to 0.3rad s1. We assume that the ocean
current is constant, with vcx = 0.2m s1, vcy = 0.2m s1.
The current is estimated using the estimator described in
(Vanni et al. (2008)). The desired speed profile is set as
vd(s) = 0.6 + 0.2 cos(0.1s). The curvature of the path is
κ(s) = 0.1 sin(0.1s).
Fig. 4.2. Trajectories of the Medusa vehicle under a con-
stant ocean current. The red-dash curves are obtained
with the Lyapunov-based controller, whereas the blue-
dash curves are obtained with the MPC scheme. The
desired path is indicated in black.
Trajectories of the vehicle initialized with different posi-
tions and orientations are depicted in Fig.4.2 while Fig.4.3
shows the performance of the Lyapnov based and the
MPC controllers. It can be observed from Fig. 4.2 that
regardless of the starting positions and orientations, the
vehicle always converges to the path with both proposed
controllers. In Fig.4.3, it is clear that the inputs satisfy
their assigned constraints (black dash lines). Similar to the
0 100 200 300
Time[s]
2
4
6
8
10
12
14
16
18
20
Path following error ||x||
0 100 200 300
Time[s]
-0.4
-0.2
0
0.2
0.4 Heading rate r
0 100 200 300
Time[s]
-1
-0.5
0
0.5
1
Speed of the virtual rabit v
Fig. 4.3. Performances of the Lyapunov-based controller
(red) and the MPC controller (blue) simulated with
realistic model of the Medusa vehicle.
kinematics simulation, we observe that the MPC controller
outperforms the Lyapunov-based controller with faster
convergence and smaller error at steady state. Overall, the
controllers show robustness against model mismatch due
to the effect of the inner loops, the vehicle dynamics, and
the error incurred in the current estimation.
5. CONCLUSION
In this paper, we have shown that the input-constrained
path following problem for autonomous marine vehicles
can be solved either with a Lyapunov-based method or
with Model Predictive Control. The main contribution of
this work lies in the fact that with the proposed con-
trollers, the path following error is globally asymptoti-
cally stable, implying that the vehicle converges to and
follows its assigned path, regardless of its initial position
and orientation. This result is not only important from
a theoretical standpoint but also makes it attractive for
practical implementations. We also shown that due to a
contractive constraint included in the proposed MPC, the
latter speeds up the convergence of the path following
error, when compared to the performance obtained with
the Lyapunov-based controller.
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Appendix A. DYNAMICS OF THE COURSE ANGLE
We recall that χ= arctan 2( ˙y, ˙x), thus
˙χ=¨y˙x˙y¨x
U2, U =p˙x2+ ˙y2.
Differentiating (1) with vr= 0, we obtain
¨x= ˙urcos ψurrsin ψ , ¨y= ˙ursin ψ+urrcos ψ.
Substituting the above in ˙χ, we obtain
˙χ=˙ur
U2kvcksin(αψ) + r
U2u2
r+urkvckcos(αψ).
(A.1)
Note that U2= (urcos ψ+vcx)2+ (ursin ψ+vcy )2.Solving
for the positive square-root of ur, we obtain
ur=−kvckcos(αψ) + m, (A.2)
where m:= qU2− kvck2sin2(αψ). Thus,
˙ur=rkvcksin(αψ)1 + kvckcos(αψ)
m+U˙
U
m.
(A.3)
Using (A.2) and (A.3), (A.1) can be simplified as
˙χ=r1kvckcos(αψ)
m˙
Ukvcksin(αψ)
Um .
... The yaw angular error is usually mapped to some predefined interval, typically −π to π radians (see e.g. [17] and [18]). At the 180 degree error point, the commanded action switches between a clockwise or counterclockwise turn. ...
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