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applied
sciences
Article
Robust Adaptive Path Following Control of an
Unmanned Surface Vessel Subject to Input Saturation
and Uncertainties
Yunsheng Fan *, Hongyun Huang and Yuanyuan Tan
School of Marine Electrical Engineering, Dalian Maritime University, Dalian 116026, China;
iamhongyun@gmail.com (H.H.); tanyuanyuan@dlmu.edu.cn (Y.T.)
*Correspondence: yunsheng@dlmu.edu.cn
Received: 17 February 2019; Accepted: 25 April 2019; Published: 1 May 2019
Abstract:
This paper investigates the path following control problem of an unmanned surface
vessel (USV) subject to input saturation and uncertainties including model parameters uncertainties
and unknown time-varying external disturbances. A nonlinear robust adaptive control scheme
is proposed to address the issue, more specifically, steering a USV to follow the desired path
at a certain velocity assignment despite the involved disturbances, by utilizing the finite-time
currents observer based line-of-sight (LOS) guidance and radial basis function neural networks
(RBFNN). Backstepping and Lyapunov’s direct method are the main design frameworks. Based on
the finite-time currents observer and adaptive control technique, an improved LOS guidance law is
proposed to obtain the desired approaching angle to the desired path, making compensations for
the effects of unknown time-varying ocean currents. Then, a kinetic controller with the capability
of uncertainties estimation and disturbances rejection is proposed based on the RBFNNs, where
the adaptive laws including leakage terms estimate the approximation error and the unknown
time-varying disturbances. Subsequently, sophisticated auxiliary control systems are employed to
handle input saturation constraints of actuators. All error signals of the closed-loop system are
proved to be locally uniformly ultimately bounded (UUB). Numerical simulations demonstrated the
effectiveness and robustness of the proposed path following control method.
Keywords:
unmanned surface vessel; path following; integral line-of-sight; finite-time currents
observer; radial basis function neural networks; input saturation.
1. Introduction
Unmanned surface vessel (USV) as an intelligent and autonomous marine equipment has received
more and more attention from the control community, for broad application in the cluttered ocean
environment, especially in cases where human intervention is not possible [
1
]. Generally speaking,
three different types of control technologies play a crucial role in the development of USVs: path
following control, trajectory tracking control and set-point control [
2
]. Many researchers propose
lots of relevant control strategies and address the issues to a various extent. This study continues
along the works and contributions of the predecessors. This paper aims at the path following control,
mainly discussing the guidance and control of a USV. Path following is usually defined as steering a
vessel to follow the desired path at a certain speed, which is not specified with temporal constraint
[
3
]. Although there are considerable theoretical studies regarding the path following and practical
engineering achievements, practical studies of the path following control for USVs have progressed
haltingly amid great difficulties. It is essential to develop a highly accurate and robust path following
controller for a USV when executing various vital missions. Therefore, under the circumstance of
severe sea state, the safe operation and mission execution can be guaranteed.
Appl. Sci. 2019,9, 1815; doi:10.3390/app9091815 www.mdpi.com/journal/applsci
Appl. Sci. 2019,9, 1815 2 of 18
Two aspects play crucial roles in the path following control scheme: guidance and control [
4
].
Guidance can refer to the popular and effective line-of-sight (LOS) guidance, refining missile guidance
approach or marine guidance [
5
,
6
]. This special guidance law exploits the geometry relationships to
generate a yaw angle known as the approaching angle, which is fed into the control system. In other
words, the control system tracks the reference yaw angle signal together with the specified velocity
tracking. Hence, the performance of the path following heavily depends on the guidance system.
It turns out that classic LOS guidance is simple and effective [
7
], albeit with limitations in the case
of being exposed to the complex ocean environment induced by waves, wind, and ocean currents.
Moreover, the traditional LOS guidance will cause large cross-tracking error when the marine surface
vehicles are in steady state, which strictly depends on the path curvature and the magnitude of the
drift force. Therefore, the traditional LOS was extended to various forms such as integral LOS (ILOS)
guidance and adaptive LOS (ALOS) guidance by Fossen. In [
8
], an ILOS guidance with integral
action is proposed to handle the constant and irrotational ocean currents and other environmental
disturbances including wind or waves. In [
9
], a new ILOS with time-varying lookahead distance is
presented with the capability of canceling the effects of constant environmental disturbances, like
constant ocean currents. In [
10
], the direct and indirect adaptive ILOS based path following controller
is proposed to deal with the ocean currents. Another way to improve the LOS guidance is to estimate
the sideslip angle caused by the external disturbance as an observer-based strategy. In [
11
], a novel
adaptive LOS (ALOS) guidance with small computation footprint is proposed where the adaptive
laws dominate the sideslip angle compensation rate. The sideslip angle is treated as an unknown
constant, which significantly limits the application of ALOS. In [
12
], a reduced-order extended state
observer (ESO) based LOS guidance is proposed to deal with the time-varying sideslip angle, which is
appropriate for straight-line and curved path following. In [
13
], the magnitude and convergence speed
of the sideslip angle is considered, which can be identified by the constructed finite-time sideslip angle
observer. However, on the one hand, it should be noted that the aforementioned literature (e.g., [
8
–
10
])
merely solve the problem in the kinematic level, i.e., ignoring the along-tracking error. On the other
hand, the method in [
10
] only adds the ocean currents in the kinematic model in terms of the relative
surge and sway velocities, whereas the along-tracking error is omitted.
The dynamics control system as the execution system of a USV is another crucial constituent
in the path following scheme. In general, USVs do not have an independent actuator in their sway
direction. This nonholonomic constraint and underactuated nature makes the control system design
much more challenging [
14
]. To satisfy the successful execution of missions and achieve expected
performance and robustness in the full range of work space from calm ocean environment to severe
ocean environment along with inaccuracy system parameters, various nonlinear control methods have
been proposed [
15
–
22
]. In [
15
], external disturbances rejection method based on the reduced-order
linear ESO is proposed. In [
16
], Do proposed a global path-tracking controller for underactuated ships
under deterministic and stochastic loads where weak nonlinearly and strong Lyapunov design method
is introduced and the estimations of disturbances are updated by projection algorithms.
In [17,18]
,
nonlinear disturbance observers are utilized to estimate the ubiquitous external disturbances in the
dynamics model of marine vehicles, where the disturbance estimation errors are proved to be UUB.
In [
19
], a nonlinear adaptive PI sliding mode tracking controller is proposed to solve the environmental
disturbance problems by relaxing the assumption of knowing the upper bound of the disturbance.
In [
20
,
21
], a novel adaptive switching-gain-based control method is proposed for a general uncertain
Euler–Lagrange system, which is not only insensitive to the nature of uncertainties but can also
alleviate the overestimation–underestimation problem. Moreover, intelligent control such as fuzzy
and neural control has been applied to deal with the uncertainties and disturbances of underactuated
marine surface ships. In [
22
], Wang proposed an adaptive online constructive fuzzy controller where
the fuzzy approximator is used to approximate the unknown disturbances. Besides, robust radial basis
function neural network control laws and iterative neural network control laws are proposed
in [23,24]
,
respectively, which are devoted to identifying and compensating the dynamical uncertainties and
Appl. Sci. 2019,9, 1815 3 of 18
external disturbances. In addition, it should be noted that few of the aforementioned path following
control schemes take into account input saturation. In fact, the practical constraint of actuators
determined by the maximum forces and moments would degrade the performance of the control
system or even make it unstable. Therefore, it is essential to implement the emulate of the constraint
on the control laws design process for reliability and robustness.
Motivated by the observations and considerations mentioned above, a finite-time currents
observer based ILOS guidance is proposed to deal with unknown time-varying ocean currents, which
is suitable for any desired parametric path with high accuracy control performance. Subsequently,
adaptive control laws based on the RBFNN are designed, which solve input saturation with
sophisticated auxiliary systems simultaneously. A robust adaptive controller is developed to address
the path following problem, which is verified to be effective via numerical simulations. The main
contributions of the paper are summarized as follows.
(1)
A finite-time currents observer based LOS guidance is presented to obtain the desired yaw angle
and estimate the unknown time-varying ocean currents precisely, which significantly influences
the performance of the control subsystem.
(2)
The RBF neural networks are incorporated into the kinetic controller to solve the uncertainties,
which does not require any prior knowledge of the dynamics of the USV and disturbances, and
the adaptive laws are designed to estimate the compound bounds of approximating errors and
external time-varying disturbances.
(3)
The input constraint effect is analyzed with auxiliary systems and the states of auxiliary systems
are utilized to make compensations for input saturation, which attenuates the challenge of
the actuators.
The rest sections are organized as follows. Section 2presents the preliminaries and problem
formulation. Section 3provides the design details of the guidance, the kinetic controller, and the
stability analysis. Simulation results are presented and analyzed in Section 4. Finally, Section 5
concludes the paper.
2. Preliminaries and Problem Formulation
2.1. RBFNN Approximation
Consider an unknown smooth nonlinear function
f(x):Rm→R
can be approximated on a
compact set Ω∈Rmby the following RBFNN:
f(x) = W∗Tϕ(x) + ε(1)
where
x∈Ω
is the input vector,
ε
is the approximation error and satisfies
|ε|6¯
ε
and
¯
ε
is a constant,
and the node number of the NN is
l>
1.
W∗∈Rl
represents the optimal weight vector, which is
defined by
W∗=arg min
ˆ
W∈Rl(sup
x∈Ωf(x)−ˆ
WTϕ(x))(2)
where
ˆ
W
is the estimation of
W∗
.
ϕ(x)=[ϕ1,ϕ2, . . . , ϕl]T:Ω→Rl
represents the radial function
vector, the element of which is chosen as the Gaussian function:
ϕi(x) = exp −(x−bi)T(x−bi)
ci2!(i=1, 2, . . . , l)(3)
where
b=[b1,b2, . . . , bl]T
and
c=[c1,c2, . . . , cl]T
are the centers of receptive field and spread of the
Gaussian function, respectively.
Appl. Sci. 2019,9, 1815 4 of 18
2.2. USV Model
As illustrated in Figure 1, the position and orientation describe the horizontal plane motion
of a USV neglecting roll, pitch, and heave, where {i} and {b} represent the inertial frame and body
fixed frame, respectively. The desired continuous path
Pd(θ) = [xd(θ),yd(θ)]T
is parameterized by a
scalar variable
θ
and
P=[x,y]T
is the position coordinate. The kinematic equations of a USV can be
described by relative velocities as follows [23]
˙
x=urcos ψ−vrsin ψ+Vx
˙
y=ursin ψ+vrcos ψ+Vy
˙
ψ=r
(4)
where
ur
and
vr
are the relative surge and sway velocities;
x
,
y
, and
ψ
express the position and
orientation in {i}; and
Vx
and
Vy
are the ocean currents represented in {b}. The dynamics of a USV is
expressed as follows [25]
˙
ur=m22
m11
vrr−d11
m11
ur−
3
∑
i=2
dui
m11 |ur|i−1ur
| {z }
fu
+1
m11
τu+1
m11
τwu
˙
vr=−m11
m22
urr−d22
m22
vr−
3
∑
i=2
dvi
m11 |vr|i−1vr
| {z }
fv
+1
m22
τwv
˙
r=(m11 −m22)
m33
urvr−d33
m33
r−
3
∑
i=2
dri
m11 |r|i−1r
| {z }
fr
+1
m33
τr+1
m33
τwr
(5)
where the positive constant parameters
mjj (j=
1, 2, 3
)
are the inertia including added mass;
dii
,
dui
,
dvi
,
and
dri (i=
2, 3
)
are the linear and quadratic hydrodynamic damping in surge, sway, and yaw;
τwu
,
τwv
, and
τwr
are the unknown time-varying environmental disturbances; and
τu
and
τr
are the
available control inputs of the surge force and the yaw moment thereby viewing it as the underactuated
control problem. Since the model parameters are directly related to the operation conditions [
26
], the
parameters of the model is uncertain. Actually, due to the physical constraint, the control inputs are
subject to nonlinear saturations, which is shown as follows
τi=
τimax,τi0>τimax
τi0,τimin 6τi06τimax
τimin,τi0<τimin
(6)
where
τimin
and
τimax (i=u
,
r)
are the minimum and maximum control inputs produced by the
actuators, referring to actual constraints of the motor’s rotational speed and rudder deflection; and
τi0
is the command control input of the path following controller.
Assumption 1. Assume that all states of a USV are measurable.
Assumption 2.
The time-varying ocean currents
ν= [Vx,Vy]T
are assumed to be irrotational and bounded,
and there exist a positive constant
M
, such that
k˙νk6M
,
M>
0. The disturbances
τwi(i=u
,
v
,
r)
are
unknown time-varying and bounded, and the first derivative of them are also bounded such that
|˙
τwi|6¯
τw
,
where ¯
τwis unknown constant.
Appl. Sci. 2019,9, 1815 5 of 18
Remark 1.
Note that in (5) the off-diagonal terms of the inertia and damping are ignored. No matter a large
scale surface vessel or a highly maneuverable unmanned surface vessel, these terms are relatively small than the
main diagonal terms. Therefore, it is reasonable to omit these terms.
Remark 2.
The external disturbances
τwu
,
τwv
, and
τwr
in Equation (5) represent the compound disturbances
of the wind and wave disturbances. The ocean currents as the form of hydrodynamic terms with relative velocities
are represented in Equation (4).
1RUWK
(DVW
^
`
L
S
J
\
^
`
E
H
\
H
[
^ `
SS
3
G
3
T
U
8
U
E
FXUUHQWV
'HVLUHG
SDWK
Figure 1. USV path following guidance information illustration.
2.3. Control Objective
In Figure 1, a local path parallel reference frame is denoted as {pp}. To arrive at{pp}, {i} should
be rotated an angle
γp(θ) = atan2 (y0d(θ),x0d(θ))
, where the notation
y0d(θ) = dyd(θ)/dθ
is used.
Therefore, the position errors can be given as follows
"xe
ye#="cos γp−sin γp
sin γpcos γp#T
| {z }
RT(γp)
(P−Pd(θ))(7)
where
xe
and
ye
are the along-tracking error and cross-tracking error, respectively, and
R(γp)
is the
rotation matrix. Meanwhile, we also have
˙
Pd(θ) = R(γp)Upp, 0T(8)
where Upp , 0Tis the velocity of {pp} with respect to {i}, represented in {pp}.
The path following control problem is concerned with designing control laws to reach and then
keep following the desired path. Once the path is reached, the vessel can maintain a desired surge
velocity assignment of
ud
. It is worth noting that a constant speed profile is frequently chosen in many
cases. Therefore, the control objective is as follows:
sup
t>0|xe|6δ1, sup
t>0|ye|6δ2, sup
t>0|ur−ud|6δ3(9)
where
δ1
,
δ2
, and
δ3
are small positive constants. Meanwhile, it is guaranteed that all error signals of
the closed-loop system are locally UUB.
Appl. Sci. 2019,9, 1815 6 of 18
Assumption 3.
The desired path should be sufficiently smooth such that its first derivative
˙
Pd
is bounded. In
addition, the desired speed assignment udand its first derivative are bounded.
3. Main Results
This section presents the design details of the path following controller to satisfy the control
objective that concluded in Section 2. The whole controller consists of two parts: the modified ILOS
guidance module and the kinetic controller, which is shown in Figure 2. The ILOS guidance is utilized
to calculate the desired yaw angle, where the constructed finite-time currents observer can provide the
fast and precise estimation of the unknown time-varying ocean currents. By incorporating the RBFNNs
into the backstepping design method, the kinetic controller are developed. Finally, the stability analysis
is presented to validate the feasibility of the proposed control approach.
ILOS guidance
External
disturbances
(wind, wave
and ocean
currents)
Yaw angle controller
Velocity controller
Yaw angle rate
controller
Unmanned
surface vessel
Kinetic controller
Desired path
G
\
G
U
X
W
U
W
G
X
U
Finite-time
currents
observer
Guidance
G G
[ \
T T
Desired
velocity Saturation
X
W
U
W
Auxiliary
system
Saturation
Auxiliary
system
X
W
'
U
W
'
RBFNNs
[ \
[ \ 9 9
\
U
X
Ö
Ö
X X
I I
Ö
Ö
[ \
9 9
Figure 2. The structure diagram of the path following controller.
3.1. Guidance
This section is devoted to designing the ILOS guidance to calculate the approaching angle with
respect to the desired path. Then, the time derivative of Equation (7) is taken to obtain the position
errors dynamics (˙
xe=urcos(ψ−γp)−vrsin(ψ−γp) + ye˙
γp+θx−Upp
˙
ye=ursin(ψ−γp) + vrcos(ψ−γp)−xe˙
γp+θy
(10)
where
θx=Vccos(βc−γp)
and
θy=Vcsin(βc−γp)
;
Vc=qV2
x+V2
y6Vcmax
and
βc=atan2(Vy,Vx)
.
By using the kinematic relationship given by Equation (8), we have
˙
θ=Upp /q˙
x2
d+˙
y2
d
. In addition, the
virtual target velocity
Upp
can be regarded as the extra degree-of-freedom to stabilize the cross-tracking
error [
5
]. To prescribe the desired yaw angle
ψd
for
ψ
, the following form of the guidance law is taken
ψd=γp−βr+atan2 −ye+αy,∆(11)
where
βr=atan2 (vr,ur)
;
∆
is the specific look-ahead distance; and
αy
is the virtual control input.
It should be noted that
αy
is introduced to shape the dynamics of the ILOS guidance by inherently
adding an integral action.
3.1.1. Estimations of Ocean Currents
Before designing the kinematic controller, the unknown ocean currents need be identified precisely.
The finite-time currents observer for Vxand Vyare designed as follows:
Appl. Sci. 2019,9, 1815 7 of 18
˙
ˆεe=Λ+g
Λ=−ρ1L1/2sig1/2 (ˆεe−εe) + ˆν
˙
ˆν=−ρ2Lsig(ˆν−Λ)
(12)
where
sigα(•) = |•|αsign(•)
;
ˆεe=[ˆ
x,ˆ
y]T
;
g= [Urcos(ψ+βr),Ursin(ψ+βr)]T
,
Ur=pu2
r+v2
r
;
ˆν= [ ˆ
Vx,ˆ
Vy]T;L=diag(l1,l2)>0; ρ1>0, ρ2>0.
Theorem 1.
Considering the proposed finite-time currents observer in Equation (12), the unknown time-varying
ocean currents νcan be precisely estimated within a finite time.
Proof. According to Equation (4), we have
˙εe=g+ν(13)
Together with the finite-time currents observer, we have
˙
˜εe=−ρ1L1/2 sig1/2(ˆεe−εe) + ˜ν
˙
˜ν=−ρ2Lsig(ˆν−Λ)−˙ν
∈ −ρ2Lsig(ˆν−Λ) + [−M,M]
(14)
where ˜εe=ˆεe−εeand ˜ν=ˆν−ν.
In light of Lemma 2 in [
27
], one can immediately have observer errors
˜εe
and
˜ν
converge to zero
within a finite time. It implies that there exists a finite time
T0
and finite constants
εb
and
vb
such that
(k˜εek<εb,k˜νk<vb,∀t<T0
˜εe≡0, ˜ν≡0, ∀t>T0
(15)
This concludes the proof.
Therefore, the parametric currents estimation errors
˜
θx=θx−ˆ
θx
and
˜
θy=θy−ˆ
θy
satisfy the
following inequality
˜
θk<q2(V2
cmax +v2
b):=¯
θ,∀t<T0
˜
θk≡0, ∀t>T0
(k=x,y)(16)
Remark 3.
With the availability of
ˆ
Vx
and
ˆ
Vy
of the finite-time currents observer, the estimations of parametric
currents ˆ
θxand ˆ
θycan be reliably obtained.
3.1.2. Design of Kinematic Controller
In this section, the kinematic control laws including the virtual control law and path variable
update law are presented.
Substituting Equation (11) into Equation (10) results in
˙
xe=Urcos ψ−γp+βr+θx−Upp +ye˙
γp
˙
ye=−Urye+αy
qye+αy2+∆2
+Urφ(ye,˜
ψ)˜
ψ+θy−xe˙
γp(17)
where
˜
ψ=ψ−ψd
and
φ(ye,˜
ψ)=sin ˜
ψ
˜
ψ
∆
q(ye+αy)2+∆2−cos ˜
ψ−1
˜
ψ
(ye+αy)
q(ye+αy)2+∆2
. Note that the upper
bound of φ(ye,˜
ψ)is 1.73.
Appl. Sci. 2019,9, 1815 8 of 18
The virtual control αyis designed to cancel θyasymptotically [10]
Urαy
qye+αy2+∆2
=ˆ
θy(18)
Solving for αyprovides a credible solution (the negative root) as follows
αy=
yeˆ
θy/Ur2−ˆ
θy/Urr∆21−ˆ
θy/Ur2+y2
e
1−ˆ
θy/Ur2(19)
In this case, the boundedness of
ˆ
θy/Ur
must be ensured such that
ˆ
θy/Ur<
1. Thus, the
boundedness of the parametric ocean currents is defined such that θy<Mθ<Ur.
The desire target velocity Upp is chosen to stabilize xeas follows
Upp =Urcos ψ−γp+βr+k1xe+ˆ
θx(20)
where
k1
is a positive design parameter. Thus, the path variable update law can be obtained according
to Equation (8):
˙
θ=Upp
q˙
x2
d+˙
y2
d
(21)
Therefore, substituting Equations (18) and (19) into Equation (17), we have
˙
xe=−k1xe+˜
θx+ye˙
γp
˙
ye=−Urye
qye+αy2+∆2
+Urφ(ye,˜
ψ)˜
ψ+˜
θy−xe˙
γp(22)
3.2. Design of Kinetic Controller
In this section, a kinetic controller is designed to track the desired approaching angle and the
desired surge velocity. Let
˜
r=r−rd
and
˜
ur=ur−ud
be the attitude tracking error and surge velocity
tracking error, respectively, where rdis the virtual control law and udis the desired surge velocity.
According to Equation (4), the time derivative of ˜
ψis given by
˙
˜
ψ=r−˙
ψd(23)
To stabilize Equation (23), the desired intermediate control law for rdis chosen as"
rd=−k2˜
ψ+˙
ψd−Urφye(24)
where k2is a positive design parameter.
From Equation (5), the dynamics of ˜
rand ˜
urare given as follows
(m33 ˙
˜
r=m33 fr+τr+τwr −m33 ˙
rd
m11 ˙
˜
ur=m11 fu+τu+τwu −m11 ˙
ud
(25)
Since the inertia and damping parameters are unknown, RBFNNs are employed to handle the
unknown parts (m33 fr−m33 ˙
rd=W∗T
1ϕ(z) + ε1
m11 fu−m11 ˙
ud=W∗T
2ϕ(z) + ε2
(26)
Appl. Sci. 2019,9, 1815 9 of 18
where
W∗
i(i=
1, 2
)
is the ideal constant weight matrix satisfying
kWi∗k6WiM
;
ϕ(z)
is the radial
basis function with
z= [ur
,
vr
,
r]T
being the input vector to the RBFNNs; and
εi(i=
1, 2
)
is the
approximation error with unknown constant upper bound such that
|εi|6¯
εi
. Denote the unknown
parts as
m33 fr−m33 ˙
rd=fu1
and
m11 fu−m11 ˙
ud=fu2
. Furthermore, there exists bounded functions
δ1and δ2such that |ε1|+|τwr|6δ1, and |ε2|+|τwu|6δ2.
Therefore, the nominal control inputs are chosen as follows by considering the input saturation
(τr0=−k3˜
r−˜
ψ−ˆ
W1ϕ(z)−ˆ
δ1h(˜
r) + kζ1σ1
τu0=−k4˜
ur−ˆ
W2ϕ(z)−ˆ
δ2h(˜
ur) + kζ2σ2
(27)
with update laws
˙
ˆ
W1=Γ1ϕ(z)˜
r−ι1ˆ
W1
˙
ˆ
δ1=ξ1˜
rh(˜
r)−λ1(ˆ
δ1−δ0
1)(28)
˙
ˆ
W2=Γ2ϕ(z)˜
ur−ι2ˆ
W2
˙
ˆ
δ2=ξ2˜
urh(˜
ur)−λ2(ˆ
δ2−δ0
2)(29)
where
k3
and
k4
are positive design parameters;
kζ1
and
kζ2
are positive design parameters;
ˆ
δi(i=
1, 2
)
is the estimation of
δi
;
h(˜
r) = tanh (˜
r/χr)
,
h(˜
ur) = tanh (˜
ur/χu)
,
χj(j=r
,
u)
is a positive constant;
Γi(i=
1, 2
)
is a positive define design matrix;
λi(i=
1, 2
)
is a small design parameter;
δ0
j(j=r
,
u)
is
the prior estimation of δj; and σi(i=1, 2)is the state of the auxiliary system.
To compensate for the constraint effects of input saturation, auxiliary dynamic systems [
28
] are
given as follows
˙
σ1=
−kσ1σ1−|˜
r∆τr|+0.5∆τr2
σ2
1
σ1+∆τr,|σ1|>µ1
0, |σ1|<µ1
(30)
and
˙
σ2=
−kσ2σ2−|˜
ur∆τu|+0.5∆τ2
u
σ2
2
σ2+∆τu,|σ2|>µ2
0, |σ2|<µ2
(31)
where
kσi(i=
1, 2
)
are positive design parameters;
µi(i=
1, 2
)
are small positive constants;
∆τr=
τr−τr0;∆τu=τu−τu0.
Therefore, the closed-loop attitude and surge velocities tracking errors dynamics become by virtue
of Equations (25) to (27)
(m33 ˙
˜
r=−k3˜
r−˜
ψ−˜
W1ϕ(z)−ˆ
δ1h(˜
r) + kς1σ1+ε1+τwr +∆τr.
m11 ˙
ue=−k4ue−˜
W2ϕ(z)−ˆ
δ2h(ue) + kς2σ2+ε2+τwu +∆τu.(32)
where
˜
Wi=ˆ
Wi−W∗
i(i=
1, 2
)
are weight matrix estimation errors; and
˜
δi=ˆ
δi−δi(i=
1, 2
)
are the
adaptive terms estimation errors.
3.3. Stability Analysis
In this section, the main theorem of the path following controller is presented.
Theorem 2.
Consider the USV model in Equations (4) and (5) in the presence of uncertainties and unknown
external time-varying disturbances under input saturation, and suppose that Assumptions 1 and 2 are satisfied,
under the guidance law in Equation (11) along with the finite-time currents observer in Equation (12). The given
path is parameterized by
θ
with the update laws in Equation (21), and the desired velocity
ud
is given as well.
The control laws in Equation (27) together with the adaptive laws in Equations (28) and (29) and the auxiliary
Appl. Sci. 2019,9, 1815 10 of 18
systems in Equations (30) and (31) are incorporated to assist in handling input saturation, guaranteeing that all
tracking error signals are locally UUB.
Proof. Consider the following Lyapunov function
V=1
2x2
e+1
2y2
e+1
2˜
ψ2+1
2m33 ˜
r2+1
2m11 ˜
u2
r+1
2
2
∑
i=1
σ2
i+1
2
2
∑
i=1
˜
WT
iΓ−1
i˜
Wi+1
2ξi
2
∑
i=1
˜
δ2
i(33)
Taking the time derivative of Equation (33) along with Equations (22)–(24) and (32) yields
˙
V6−k1x2
e−Ur
q(ye+αy)2+∆2
y2
e−k2˜
ψ2−k3˜
r2−˜
W1ϕ(z)˜
r−ˆ
δ1h(˜
r)˜
r
+kζ1σ1˜
r+δ1˜
r+∆τr˜
r−k4˜
u2
r−˜
W2ϕ(z)˜
ur−ˆ
δ2h(˜
ur)˜
ur+kζ2σ2˜
ur+δ2˜
ur+∆τu˜
ur
+˜
θxxe+˜
θyye+
2
∑
i=1
σi˙
σi+
2
∑
i=1
˜
WT
iΓ−1
i˙
ˆ
Wi+1
ξi
2
∑
i=1
˜
δi˙
ˆ
δi
(34)
(1) When
|σi|>µi(i=
1, 2
)
, according to Equations (27) to (30) and Young’s equalities
kζ1σ1˜
r6
1
2˜
r2+1
2k2
ζ1σ2
1
,
σ1∆τr61
2σ2
1+1
2∆τ2
r
,
kζ2σ2˜
ur61
2˜
u2
r+1
2k2
ζ2σ2
2
,
σ2∆τu61
2σ2
2+1
2∆τ2
u
,
˜
θxxe61
2¯
θ2+1
2x2
e
,
and ˜
θyye6k1
2¯
θ2+1
2k1y2
e, we have
˙
V6−k1−1
2x2
e− Ur
py2
e+∆2−1
2k1!y2
e−k2˜
ψ2−kσ1−1
2−1
2k2
ζ1σ2
1−k3−1
2˜
r2
−kσ2−1
2−1
2k2
ζ2σ2
2−k4−1
2˜
u2
r−
2
∑
i=1
ιi˜
WT
iˆ
Wi−
2
∑
i=1
λi˜
δiˆ
δi−δ0
i
+˜
rδ1−ˆ
δ1h(˜
r)+˜
δ1h(˜
r)˜
r+˜
urδ2−ˆ
δ2h(˜
ur)+˜
δ2h(˜
ur)˜
ur+1+k1
2¯
θ2
(35)
Consider the following inequality of the hyperbolic tangent function holds for any
χ>
0 and for
any v∈R[29]
06|v|−vtan v
χ6κχχ(36)
where κχis a constant that satisfies κχ=e−(κχ+1), i.e., κχ=0.2785.
It is worth noting that the following equalities hold
−
2
∑
i=1
ιi˜
WT
iˆ
Wi6−
2
∑
i=1
ιi
2˜
WT
i˜
Wi+
2
∑
i=1
ιi
2kW∗
ik2(37)
−
2
∑
i=1
λi˜
δiˆ
δi−δ0
i6−
2
∑
i=1
λi
2˜
δ2
i+
2
∑
i=1
λi
2δi−δ0
i2(38)
˜
rδ1−ˆ
δ1tan ˜
r
χr+˜
δ1tan ˜
r
χr˜
r6δ1|˜
r|−˜
rtan ˜
r
χr60.2785χrδ1(39)
˜
urδ2−ˆ
δ2tan ˜
ur
χu+˜
δ2tan ˜
ur
χu˜
ur6δ2|˜
ur|−˜
urtan ˜
ur
χu60.2785χuδ2(40)
Appl. Sci. 2019,9, 1815 11 of 18
Substituting Equations (36)–(39) into Equation (35), we have
˙
V6−k1−1
2x2
e− Ur
py2
e+∆2−1
2k1!y2
e−k2˜
ψ2−k3−1
2˜
r2−k4−1
2˜
u2
r
−
2
∑
i=1kσi−1
2−1
2k2
ζiσ2
i−
2
∑
i=1
ιi
2˜
WT
i˜
Wi−
2
∑
i=1
λi
2˜
δ2
i+
2
∑
i=1
1
2δi−δ0
i2
+
2
∑
i=1
ιi
2WiM2+0.2785 (χrδ1+χuδ2)+1+k1
2¯
θ2
6−κ1V+ϑ1
(41)
where
κ1=min 2k1−1, 2Ur
√y2
e+∆2−1
k1, 2k2,2k3−1
m33 ,2k4−1
m11 , min
i=1,2 2kσi−1−k2
ζi, min
i=1,2 (ιiλmin(Γi)), min
i=1,2 (λiξi)
,
k1=max 1
2,√y2
e+∆2
2Ur
,
λmin(•)
denotes the minimum eigenvalue of a matrix, and
ϑ1=2
∑
i=1
1
2δi−δ0
i2+2
∑
i=1
ιi
2WiM2+0.2785 (χrδ1+χuδ2)+1+k1
2¯
θ2.
(2) When
|σi|<µi(i=
1, 2
)
, we have
2
∑
i=1
σi˙
σi=0
. According to Equations (30) and (31) and
the inequalities
2
∑
i=1
1
2k2
ζiσ2
i6−2
∑
i=1
1
2k2
ζiσ2
i+2
∑
i=1
k2
ζiµ2
i
,
˜
r∆τr61
2˜
r2+1
2∆τ2
r
, and
˜
ur∆τu61
2˜
u2
r+1
2∆τ2
u
,
Equation (34) becomes
˙
V6−k1−1
2x2
e− Ur
py2
e+∆2−1
2k1!y2
e−k2˜
ψ2−(k3−1)˜
r2−(k4−1)˜
u2
r
−
2
∑
i=1
k2
ζi
2σ2
i−
2
∑
i=1
ιi
2˜
WT
i˜
Wi−
2
∑
i=1
λi
2˜
δ2
i+
2
∑
i=1
1
2δi−δ0
i2+
2
∑
i=1
ιi
2WiM2
+0.2785 (χrδ1+χuδ2)+
2
∑
i=1
k2
ζiµ2
i+1
2∆τ2
r+1
2∆τ2
u+1+k1
2¯
θ2
6−κ2V+ϑ2
(42)
where
κ2=min 2k1−1, 2Ur
√y2
e+∆2−1
k1, 2k2,2k3−2
m33 ,2k4−2
m11 , min
i=1,2 k2
ζi, min
i=1,2 (ιiλmin(Γi)), min
i=1,2 (λiξi)
,
k1=max 1
2,√y2
e+∆2
2Ur
;
ϑ2=2
∑
i=1
1
2δi−δ0
i2+2
∑
i=1
ιi
2WiM2+
0.2785
(χrδ1+χuδ2)+2
∑
i=1
k2
ζiµ2
i+1
2∆τ2
r+
1
2∆τ2
u+1+k1
2¯
θ2.
Synthesizing Equations (41) and (42), we have
˙
V6−κV+ϑ(43)
where
κ=min{κ1
,
κ2}
and
ϑ=max{ϑ1
,
ϑ2}
with the design parameters satisfying the conditions:
k1>max 1
2,√y2
e+∆2
2Ur
,
k2>
0,
k3>
1,
k4>
1,
kζ1>
0,
kζ2>
0,
kσ1>1
2k2
ζ1+1
2
,
kσ2>1
2k2
ζ2+1
2
. Then,
the following inequality can be obtained
06V6V(0)−ϑ
κe−κt+ϑ
κ(44)
In conclusion, it follows the definition of
V
that the tracking error signals
ηe=
[xe,ye,˜
ψ,˜
r,˜
ur]T
are locally UUB, which ultimately converges to the compact sets
Ω1=
ηe∈R5|kηek6r2V(0)−ϑ
κe−κt+2ϑ
κ
. The ultimate compact set can be easily tuned by
Appl. Sci. 2019,9, 1815 12 of 18
adjusting the design parameters. Meanwhile, the parameters estimation errors
˜
W1
,
˜
W2
,
˜
δ1
, and
˜
δ2
are
locally UUB. Theorem 2 is thus proved.
3.4. Sway Dynamics
For the sway velocity dynamics, considering the Lyapunov function
Vv=1
2v2
r
and taking the
time derivative of it based on Equation (5) yields
˙
Vv=vr˙
vr
=−m11
m22
urrvr−d22
m22
v2
r−dv2
m11 |vr|v2
r−dv3
m11
v4
r+1
m22
τwv
6−χvr+1
m22
¯
τw
(45)
where
χ=m11
m22 urr
. Based on the above analyses, the boundedness of
˜
ur
and
˜
r
is guaranteed, thus
χ
is
bounded. It should be noted that
dvi
and
mii (i=
2, 3
)
are positive constants. According to Krstic, M.;
Kanellakopoulos, I.; Kokotovic, P. V. [30], we have
Vv6Vv(t0)e−χ(t−t0)+¯
τw1−e−χ(t−t0)
m22χ
6Vv(t0) + ¯
τw
m22χ
(46)
Therefore, the boundedness of the sway velocity vris guaranteed.
4. Simulations
The effectiveness and robustness of the proposed path following control method were evaluated
based on the platform of MATLAB. The Cybership II [
31
] was taken as the control object whose
parameters were as follows:
m11 =
25.8,
m22 =
33.8,
m33 =
2.76,
d11 =
0.9257,
d22 =
2.8909,
d33 =
0.5.
For simplicity, we ignored the off-diagonal terms of the inertia and damping. The maximum actuated
force and moment were 2
N
and 1.5
Nm
. From the beginning of the simulations, the ocean currents
with time-varying speed were given as
Vx=
0.08
sin (0.1t)m/s
and
Vy=
0.04
sin(
0.1
t)m/s
. The
time-varying external disturbances were generated with the first-order Markov process
˙
τwu +ς1τwu =
w1
,
˙
τwv +ς2τwv =w2
,
˙
τwr +ς3τwr =w3
, where
wi
and
ςi(i=
1, 2, 3
)
are zero-mean Gaussian white
noise and constants, respectively [
32
]. The parameters of the controller are listed in Table 1. The node
numbers and widths of RBFNNs were chosen as: node number
l=
21 and the widths
bi=
3
(i=
1, 2,
. . .
,
l)
. The neural active region was chosen as
[|u|,|v|,|r|]∈[[0, 2],[0, 1.5],[0, 1.5]]T
. Performance
comparisons between the proposed finite-time currents observer based ILOS guidance with adaptive
RBFNN (FCONN) controller and the indirect adaptive observer based [
10
] ILOS guidance with
adaptive RBFNN (IAONN) controller are presented in the following two control scenarios.
Table 1. Parameters of the path following controller.
Notation Value Natation Value Natation Value Natation Value Natation Value
k11l24Γ1ii(i=21)500 ξ220 χr0.01
k22ρ10.01 Γ2ii(i=21)50 λ10.01 χu1
k34ρ20.03 ι10.05 λ20.01 ∆2.51
k45kζ11.2 ι20.05 δ0
10.1
l1100 kζ21.2 ξ150 δ0
10.1
Case 1: The desired path and speed assignment were chosen as
Pd=[θ,θ]T
and
ud=
0.5
m/s
.
The initial states were given as [x(0),y(0),ur(0),vr(0),r]T=[0 m, 2 m, 0.01 m/s, 0 m/s, 0 rad/s]T.
Appl. Sci. 2019,9, 1815 13 of 18
The simulation results are shown in Figure 3a–g. Table 2summarizes the performance indices
based on the integrated absolute error (IAE) and the time integrated absolute error (ITAE), which were
used to evaluate the transient performance and steady-state performance. The proposed FCONN
control method could drive the USV following the desired path with a high-precision and better
transient process (Figure 3a). Their detailed distinction is more clearly shown in Figure 3b,c, specifically
the smaller along- and cross-tracking errors and the smaller heading and surge velocity tracking errors.
Meanwhile, the lower IAE and ITAE metrical values of the cross-tracking error revealed the better
transient and steady-state performance. Figure 3d shows that the proposed finite-time currents
observer could identify the time-varying currents accurately, whereas the IAONN control scheme had
an obvious oscillation during the transient process, as well as lower accuracy in the steady state. Figure
3d demonstrates that the RBFNNs could capture the unknown dynamical uncertainties precisely and
Figure 3e presents the compound bounds and their estimation, which played decisive roles in driving
the dynamics state
r
and
ur
to their real value. Figure 3f depicts the profile of the control inputs where
the input saturation (IS) problem Was effectively compensated by the auxiliary system. The control
inputs of the proposed method were in the specified region. Hence, in the case, the proposed FCONN
path following control method was more effective and robust according to these simulation results.
0 10 20 30 40 50 60 70 80
0
10
20
30
40
50
60
70
80
y[m]
x[m]
desired path FCONN ICONN
2 3 4
0
2
4
(a) line path following.
0 50 100 150 200
−0.5
0
0.5
1
1.5
xe[m]
FCONN ICONN
0 50 100 150 200
−0.5
0
0.5
1
1.5
t[s]
ye[m]
FCONN ICONN
60 80 100
−0.02
0
0.02
(b) Along- and cross-tracking error in Case 1.
0 50 100 150 200
−0.3
−0.2
−0.1
0
˜
ψ[rad]
FCONN ICONN
0 50 100 150 200
−0.6
−0.4
−0.2
0
0.2
t[s]
˜u[ m/s]
FCONN ICONN
(c) Profile of the heading and surge velocity tracking in Case 1.
0 50 100 150 200
−0.1
0
0.1
0.2
0.3
θx,
ˆ
θx
parametric currents FCONN ICONN
0 50 100 150 200
−0.1
0
0.1
0.2
0.3
t[s]
θy,
ˆ
θy
parametric currents FCONN ICONN
(d) Estimations of parametric currents in Case 1.
Figure 3. Cont.
Appl. Sci. 2019,9, 1815 14 of 18
0 50 100 150 200
−1
0
1
2
fu1 ˆ
fu1
0 50 100 150 200
−100
−50
0
50
t[s]
fu2 ˆ
fu2
60 80 100
−1
−0.5
0
0.5
(e) Estimations of dynamical uncertainties using RBFNN in
Case 1.
0 50 100 150 200
−2
−1
0
1
[Nm]
δ1ˆ
δ1
0 50 100 150 200
−50
0
50
100
t[s]
[N]
δ2ˆ
δ2
(f) Estimations of compound bound in Case 1.
0 50 100 150 200
0
50
100
τr[Nm]
Without IS With IS
0 50 100 150 200
−1
0
1
2
3
4
t[s]
τu[N]
Without IS With IS
0 1 2 3 4 5
−2
0
2
(g) Profile of the control inputs in Case 1.
Figure 3. Line path following results.
Table 2. Performance indices in these two path following scenarios.
Control Law
Line Path Following Curvilinear Path Following
IAE(·103)
Rt
0|ye|dτ
ITAE(·103)
Rt
0t|ye|dτ
IAE(·104)
Rt
0|ye|dτ
ITAE(·104)
Rt
0t|ye|dτ
FCONN 1.92 5.78 2.58 9.01
IAONN 4.18 13.12 7.32 20.32
Case 2: Similar to Case 1, another comparison is presented to verify the performance in the
case of following a curvilinear path with the same design parameters. The desired path and speed
assignment were chosen as
Pd=θ,[10sin(0.1θ)]T
and
ud=
0.5
m/s
. The initial states were given as
[x(0),y(0),ur(0),vr(0),r]T=[0 m, 2 m, 0.01 m/s, 0 m/s, 0 rad/s]T
. The simulations results are shown
in Figure 4a–g and the performance quantification indices are summarized in Table 2. As illustrated in
Figure 4a, although following the curvilinear path, the proposed FCONN method behaved almost the
same in both control scenarios. As shown in Figure 4b, the along- and cross-tracking errors oscillated
to varying degrees for the poor performance of IAONN, whereas the position errors of FCONN could
smoothly and steadily converge to a small neighborhood around zero within a short time. Moreover,
the smaller IAE and ITAE metrical values verified it. Figure 4c shows the slight oscillation of the
heading and surge velocity tracking error of IAONN. Moreover, in Figure 4d, the poor performance
of estimating ocean currents of IAONN undoubtedly degraded the tracking performance. Figure
4e,f shows the exceptional performance of the dynamical uncertainties estimation and disturbance
Appl. Sci. 2019,9, 1815 15 of 18
rejection of the FCONN method. In addition, the control inputs were in the specified region for
introducing the auxiliary system, as depicted in Figure 4g. Overall, the proposed control method
achieved satisfactory performance and robustness in both cases with fast and accurate estimations of
the unknown time-varying ocean currents and dynamical uncertainties, and satisfactory rejection of
external disturbances.
0 10 20 30 40 50 60 70
0
10
20
30
40
50
60
70
y[m]
x[m]
desired path FCONN ICONN
0 2 4
0
5
10
(a) Curvilinear path following.
0 50 100 150 200
−0.5
0
0.5
1
xe[m]
FCONN ICONN
0 50 100 150 200
−0.5
0
0.5
1
1.5
2
t[s]
ye[m]
FCONN ICONN
60 80 100
−0.01
0
0.01
100 120 140
−5
0
5x 10−3
(b) Along- and cross-tracking error in Case 2.
0 50 100 150 200
−2
−1
0
1
2
˜
ψ[rad]
FCONN ICONN
0 50 100 150 200
−0.6
−0.4
−0.2
0
0.2
t[s]
˜u[m/s]
FCONN ICONN
60 80 100
−0.02
0
0.02
(c) Profile of the heading and surge velocity tracking in Case 2.
0 50 100 150 200
−0.1
0
0.1
0.2
0.3
θx,
ˆ
θx
parametric currents FCONN ICONN
0 50 100 150 200
−0.1
−0.05
0
0.05
0.1
t[s]
θy,
ˆ
θy
parametric currents FCONN ICONN
(d) Estimations of parametric currents in Case 2.
0 50 100 150 200
−30
−20
−10
0
10
fu1 ˆ
fu1
0 50 100 150 200
−80
−60
−40
−20
0
20
t[s]
fu2 ˆ
fu2
60 80 100
−1
0
1
fu1 ˆ
fu1
60 80 100
−2
−1
0
1
fu2 ˆ
fu2
(e) Estimations of dynamical uncertainties using RBFNN in
Case 2.
0 50 100 150 200
−10
0
10
20
[Nm]
δ1ˆ
δ1
0 50 100 150 200
−50
0
50
100
t[s]
[N]
δ2ˆ
δ2
(f) Estimations of compound bound in Case 2.
Figure 4. Cont.
Appl. Sci. 2019,9, 1815 16 of 18
0 50 100 150 200
−10
0
10
20
τr[Nm]
Without IS With IS
0 50 100 150 200
−2
0
2
4
t[s]
τu[N]
Without IS With IS
024
−10
0
10
20
Without IS With IS
(g) Profile of the control inputs in Case 2.
Figure 4. Curvilinear path following results.
5. Conclusions
In this paper, a path following control scheme for a USV subject to input saturation and
uncertainties has been proposed by resorting to the finite-time currents observer based ILOS guidance,
the adaptive RBFNN, and the auxiliary dynamic system. The finite-time currents observer based
ILOS guidance is applied to obtain the desired yaw angle, where the incorporated finite time
currents observer can provide the precise estimations of the unknown time-varying ocean currents.
Simultaneously, the RBF neural networks and the adaptive laws with leakages terms can provide the
precise estimations of dynamical uncertainties and the compound bounds of the approximation errors
and external disturbances, without knowing any prior knowledge of the time-varying disturbance.
The auxiliary control system is introduced to handle input saturation of the actuators. It has been
proved that all error signals of the closed-loop system are locally UUB. Finally, both linear and curved
path following are presented and compared with the preceding control method. Simulations results
have verified that the proposed control method can achieve satisfactory performance and robustness.
Future work will cover the aspect of the position error constraint to ensure that the USV can work
in these situations including the narrow passage and the channel between obstacles, as well as the
precise estimation of the sideslip angle.
Author Contributions:
F.Y., H.H. and Y.T. conceived the framework and wrote the paper. F.Y. and H.H. designed
the controller and H.H. and Y.T. were responsible for the formula analyses. H.H. made the simulations and Y.T.
analyzed the data. F.Y., H.H. and Y.T. discussed the results and contributed to the whole manuscript.
Funding:
This research was funded by National Nature Science Foundation under grant number 51609033, in part
by Natural Science Foundation of Liaoning Province under Grant number 201801732, and in part by Fundamental
Research Funds for the Central Universities under Grant number 3132016312.
Conflicts of Interest: The authors declare no conflict of interest.
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