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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]T∈R3

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 ]T∈R2expressed in the inertial frame {I}. The

3-DOF kinematic model of the vehicle is described by

˙x=urcos ψ−vrsin 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=r1−kvckcos(α−ψ)

m−˙

Ukvcksin(α−ψ)

Um ,

(2b)

where rχdenotes the course rate, α= arctan 2(vcy , vcx),

and m=qU2− kvck2sin2(α−ψ) (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 ≤U≤Umax,|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 (α−ψ), 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=χ−ψF. 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)

rχ−κ(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)ψe−k2tanh(ψe) + κ(s)v

(5)

solves the input constrained-path following problem, where

k1, k2, k ∈R>0are tuning parameters that satisfy

0< k1≤vmax −max(vd(s)),

0.5kmax(vd(s)) + k2≤rχmax −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)/ψe≤1 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 rχ≤rχmax.

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 := {x∈R3|ex= 0, ψe= 0}. In Ω,

since lim

ψe→0sin ψe/ψe= 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)

rχ−κ(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:R3→R≥0and its as-

sociated stabilizing constrained control law un:R3→U.

Finally, l:R3×U×R→R≥0is 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=iδ,i∈N+, 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, rχ−κ(s)v]T=0.

A2.2 Given the path following error dynamics in (8), there

exists a global Lyapunov function V:R3→R≥0

such that Vis positive deﬁnite and V(x) = 0 if

and only if x=0and an associated nonlinear

feedback control law un:R3→Uthat 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

ua→0once 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), Q∈R3×3denotes a diagonal positive deﬁnite

matrix that penalizes the path following error, whereas

R∈R2×2is 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)m−1. The vehicle inputs are constrained with

0≤U≤1.2m s−1and |r| ≤ 0.5rad s−1. The speed proﬁle

along a path is speciﬁed as vd(s) = 0.8+0.2 cos(0.1s)m s−1.

Thus, we obtain max(vd(s)) = 1ms−1, max(κ(s)) =

0.1m−1, rχmax = 0.5rad s−1. We set vmax = 1.5m s−1,

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 s−1, and the references for heading rate

from -0.3rad s−1to 0.3rad s−1. We assume that the ocean

current is constant, with vcx = 0.2m s−1, vcy = 0.2m s−1.

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 ψ−urrsin ψ , ¨y= ˙ursin ψ+urrcos ψ.

Substituting the above in ˙χ, we obtain

˙χ=−˙ur

U2kvcksin(α−ψ) + r

U2u2

r+urkvckcos(α−ψ).

(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(α−ψ) + m, (A.2)

where m:= qU2− kvck2sin2(α−ψ). Thus,

˙ur=rkvcksin(α−ψ)−1 + kvckcos(α−ψ)

m+U˙

U

m.

(A.3)

Using (A.2) and (A.3), (A.1) can be simpliﬁed as

˙χ=r1−kvckcos(α−ψ)

m−˙

Ukvcksin(α−ψ)

Um .