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Ocean Engineering

journal homepage: www.elsevier.com/locate/oceaneng

Fuzzy unknown observer-based robust adaptive path following control ofunderactuated surface vehicles subject to multiple unknowns☆

Ning Wanga,∗, Zhuo Suna, Jianchuan Yinb, Zaojian Zouc, Shun-Feng Sud

a Center for Intelligent Marine Vehicles and School of Marine Electrical Engineering, Dalian Maritime University, Dalian, 116026, PR ChinabNavigation College, Dalian Maritime University, Dalian, 116026, PR Chinac Department of Naval Architecture and Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200030, PR ChinadDepartment of Electrical Engineering, National Taiwan University of Science and Technology, Taiwan, PR China

A R T I C L E I N F O

Keywords:Surge-guided line-of-sight guidancePath following controlUnderactuated surface vehicleFuzzy unknown observer

A B S T R A C T

In this paper, suffering from multiple unknowns including unmodeled dynamics, uncertainties, and disturbances,a fuzzy unknown observer-based robust adaptive path following control (FUO-RAPFC) scheme for an under-actuated surface vehicle (USV) is proposed. Main contributions are as follows: (1) A surge-guided line-of-sight(SGLOS) guidance law is created to guide surge speed and heading angle, simultaneously, and thereby adaptingto path following errors to enhance robustness and flexibility; (2) By virtue of combining auxiliary observationdynamics with adaptive approximation error compensation, the FUO is innovatively devised to accuratelylumped unknowns rather than bounded observations; (3) FUO-based robust adaptive tracking control laws areeventually synthesized and ensure that the guided signals can be globally asymptotically tracked. Simulationstudies are conducted to demonstrate remarkable performance of the FUO-RAPFC scheme.

1. Introduction

The application of underactuated surface vehicles (USVs) becomesmore and more prevalent due to its flexibility and versatility in bothmaritime civilian and military (Wynn and Huvenne, 2014; Xu and Xiao,2007; Xiang et al., 2016; Wang et al., 2018a, 2018b). Accordingly,motion control of USVs has been being within a research hotspot(Allotta et al., 2016; Mai et al., 2017; Zhao et al., 2016). In the litera-ture, among most of scenarios including stabilization, trajectorytracking and path following, it is of practical importance to follow apredefined path without time constraint, and thereby resulting thatpath following control (PFC) takes a critical part in various high-riskunmanned maritime tasks.

The overall PFC scheme of a USV usually includes guidance moduleand control module (Fossen, 1994; Fossen et al., 2003; Breivik andFossen, ), whereby the former is utilized to generate guidance signalswhich may include attitudes and/or velocities and the later is requiredto follow previous guided references. The former guidance module

works at guiding attitudes and/or velocities. The line-of-sight (LOS)guidance method has been prevalently used for path following of aUSV, whereby an LOS projection algorithm was deployed to follow asegmented path actually connected by waypoints (Fossen et al., 2003).The proportional line-of-sight (PLOS) guidance approach was proposedin Pettersen and Lefeber (2001) and Fossen and Pettersen (2014),where the guided heading angle was derived from the arctangent valueassociated with cross-track error and lookahead distance. However, thesideslip angle, i.e., the discrepancy between the heading and the courseangles, which may be caused by multiple unknowns and/or maneuvers,has not been addressed in previous works. In Brhaug et al. (2008) andCaharija and Pettersen (2016), taking sideslip angle as a slow time-varying unknown term, the integral line-of-sight (ILOS) guidance wasproposed to compensate the unknown sideslip. The adaptive LOS(ALOS) guidance method was proposed in Fossen et al. (2015), wherebyunknown sideslip was estimated by an adaptive term. The guidanceapproach for curved paths was developed in Moe et al. (2016). It shouldbe noted that the previous guidance laws proposed in Fossen et al.

https://doi.org/10.1016/j.oceaneng.2019.02.017Received 17 May 2018; Received in revised form 13 January 2019; Accepted 3 February 2019

☆ This work is supported by the National Natural Science Foundation of P. R. China (under Grants 51009017, 51379002 and 61803063), the Fund for DalianDistinguished Young Scholars (under Grant 2016RJ10), the Fund for Liaoning Innovative Talents in Colleges and Universities (under Grant LR2017024), the StableSupporting Fund of Science and Technology on Underwater Vehicle Laboratory (SXJQR2018WDKT03), and the Fundamental Research Funds for the CentralUniversities (under Grants 3132016314 and 3132018126).

∗ Corresponding author.E-mail addresses: n.wang.dmu.cn@gmail.com (N. Wang), 18042684439@163.com (Z. Sun), yinjianchuan@dlmu.edu.cn (J. Yin), zjzou@sjtu.edu.cn (Z. Zou),

sfsu@mail.ntust.edu.tw (S.-F. Su).

Ocean Engineering 176 (2019) 57–64

Available online 22 February 20190029-8018/ © 2019 Elsevier Ltd. All rights reserved.

T

(2003); Breivik and Fossen, ; Pettersen and Lefeber (2001); Fossen andPettersen (2014); Brhaug et al. (2008); Caharija and Pettersen (2016);Fossen et al. (2015); Moe et al. (2016) inevitably suffer from the sin-gularity problem in the selection of projection points. Nevertheless,surge velocity is roughly treated as a user-defined parameter, and isusually predefined as a constant. In this context, the USV is actuallycontrolled only by the rudder torque, and thereby not only cuttingdown the overall manipulability but also increasing maneuveringburden on the rudder.

The control module within the path following scheme is desired torender the USV track the guided signals as accurately as possible. In thiscontext, nonlinear tracking controllers can be directly derived frombackstepping-like techniques (Wang et al., 2017a, 2017b; Zhou et al.,2017a; Li et al., 2017). In Li et al. (2008), the adaptive controllers weredesigned to solve waypoint tracking problem of a USV, whereby alltracking error signals within the closed-loop system were rendered tobe uniformly ultimately bounded. In Miao et al. (2017), the auto-dis-turbances rejection control (ADRC) approach was deployed to achievePFC of a UUV. Recently, an adaptive robust finite-time tracking controlschemes using non smooth and homogeneity tools for trajectorytracking of a marine surface vehicle with unknown disturbances havebeen proposed in Wang et al. (2016a); Liu et al. (2017). A singularperturbation control strategy has been developed in Yi et al. (2016). Inorder to handle complex unknowns, finite-time observer based motioncontrol schemes for marine vehicles have been addressed in (Wanget al., 2017d. Note that intelligent approaches have been extensivelyapplied to various industrial areas (Wang et al., 2018d, 2019). Fuzzysystems (Wang et al., 2017e, 2018e), artificial neural networks (Wanget al., 2015a), and fuzzy neural networks (Wang and Er, 2015), etc,have been intensively explored within trajectory tracking control ofmarine surface vehicles. In Wang et al. (2017f), an adaptive universe-based fuzzy control scheme with retractable fuzzy partitioning in globaluniverse of discourse has been created to achieve global asymptoticmodel-free trajectory independent tracking. However, comparing totracking control approaches of fully actuated marine vehicles, complexunknowns pertaining to underactuated marine vehicles are still hardlybe tackled sufficiently due to underactuation appearing in sway dy-namics.

In this paper, motivated by above observations, a fuzzy unknownobserver based robust adaptive path following control (FUO-RAPFC)scheme is created to achieve accurate path following of a USV withmultiple unknowns including internal unmodeled dynamics, e.g.,parametric unknowns/uncertainties, and external disturbances, si-multaneously, which have not been addressed in previous works.Stemming from line-of-sight (LOS) approach, a surge-guided LOS(SGLOS) guidance law is devised and ensures that the reference surgespeed can be autonomously guided with respect to path following er-rors. To be more interesting, by designing auxiliary observation dy-namics, the FUO is innovatively devised to accurately estimate multipleunknowns, and thereby contributing to robust adaptive tracking con-trollers which guarantee that guided signals can be accurately tracked.

The rest is organized as follows. Section 2 states problem formula-tion. The SGLOS guidance law and the FUO-based robust adaptivetracking controllers are presented in Sections 3 and 4, respectively.Simulation studies are conducted in Section 5. Section 6 concludes thispaper.

2. Problem formulation

2.1. USV kinematics and dynamics

The kinematics of a USV are as follows (Fossen):

= −= +=

x u ψ v ψy u ψ v ψψ r

˙ cos sin˙ sin cos˙ (1)

where x y ψ( , , ) are planar position and heading orientation of the USVin the earth-fixed frame, and u v r( , , ) are surge, sway and yaw velocitieswithin the body-fixed frame, respectively.

Suffering from the internal unmodelled dynamics, parametric per-turbations, and external environmental disturbances, the dynamics of aUSV can be formulated as follows:

+ = + − + + ++ = − + − + ++ = − + − − +

+ +

m m u m m vr d d u τ τm m v m m ur d d v τm m r m m m m uv d d r

τ τ

( Δ ) ˙ ( Δ ) ( Δ )( Δ ) ˙ ( Δ ) ( Δ )( Δ ) ˙ [ (Δ Δ )] ( Δ )

u δ

δ

r δ

11 11 22 22 11 11

22 22 11 11 22 22

33 33 11 22 11 22 33 33

u

v

r (2)

where =m i( 1,2,3)ii and =d i( 1,2,3)ii denote ship inertia includingadded mass and the hydrodynamic damping in surge, sway and yaw,

=m iΔ ( 1,2,3)ii and =d iΔ ( 1,2,3)ii represent parametric perturbations.The available controls are surge control force τu and the yaw controlmoment τr , and τ τ τ( , , )δ δ δu v r are external disturbances in surge, sway andyaw directions. The dynamics (2) can be further rearranged as follows:

= + +

= +

= + +

u f u v r

v f u v r

r f u v r

˙ ( , , )

˙ ( , , )

˙ ( , , )

uτ

mD u v r

m

vD u v r

m

rτ

mD u v r

m

( , , )

( , , )

( , , )

u u

v

r r

11 11

22

33 33 (3)

where

= −

= − −

= −−

f u v r vr u

f u v r ur v

f u v r uv r

( , , )

( , , )

( , , )

umm

dm

vmm

dm

rm m

mdm

( )

2211

1111

1122

2222

11 2233

3333 (4)

and Du, Dv, Dr are treated as lumped unknowns including both un-modeled dynamics, uncertainties and disturbances, and are given by:

= − + − += − − + −= − + − − +

D u v r m u m vr d u τD u v r m v m ur d v τD u v r m r m m uv d r τ

( , , ) Δ ˙ Δ Δ( , , ) Δ ˙ Δ Δ( , , ) Δ ˙ (Δ Δ ) Δ

u δ

v δ

r δ

11 22 11

22 11 22

33 11 22 33

u

v

r (5)

Remark 1. It should be noted that lumped unknowns Du, Dv and Dr areessentially associated with USV dynamics, parametric uncertainties anddisturbances. Apparently, the foregoing unknowns can impossibly beapproximated accurately by any nonlinear mapping since both internalderivatives and external disturbances exist.

2.2. Path following error dynamics

In Fig. 1, a reference curve path that is parameterized by a time-dependent variable ϖ t( ). The path-tangent reference frame (PTRF) atthe point (x ϖ y ϖ( ), ( )p p ) is rotated with ϕp relative to the inertial co-ordinate system given by

= ′ ′ϕ y ϖ x ϖatan2( ( ), ( ))p p p (6)

where ′ = ∂ ∂x ϖ x ϖ( ) /p p , and ′ = ∂ ∂y ϖ y ϖ( ) /p p .The path following errors between x y( , ) and x y( , )p p expressed in the

PTRF frame are as follows:

⎡⎣⎢

⎤⎦⎥

= ⎡

⎣⎢

− ⎤

⎦⎥

⎡⎣⎢

−−

⎤⎦⎥

xy

ϕ ϕϕ ϕ

x xy y

cos sinsin cos

e

e

p p

p p

Tp

p (7)

where xe and ye are referred to the cross- and along-track errors, re-spectively.

Accordingly, tracking error dynamics can be obtained

= − − − + −

= − + − −

x u ψ ϕ v ψ ϕ ϕ y u

y u ψ ϕ v ψ ϕ ϕ x

˙ cos( ) sin( ) ˙

˙ sin( ) cos( ) ˙e p p p e tar

e p p p e (8)

where utar is the total speed of the “virtual target” as follows:

N. Wang, et al. Ocean Engineering 176 (2019) 57–64

58

= ′ + ′u ϖ x ϖ y ϖ˙ ( ) ( )p ptar2 2

(9)

By defining the sideslip angle as follows:

=β atan v u2( , ) (10)

yields tracking error dynamics given by

= − − − + −

= − + − −

x u ψ ϕ u ψ ϕ β ϕ y u

y u ψ ϕ u ψ ϕ β ϕ x

˙ cos( ) sin( )tan ˙

˙ sin( ) cos( )tan ˙e p p p e tar

e p p p e (11)

where sideslip angle is the actually unknown.

3. Surge-guided LOS (SGLOS) guidance

As shown in Fig. 2, a surge-guided line-of-sight (SGLOS) guidancescheme consisting of surge, heading and virtual target guidance laws isdesigned in guidance system.

Within the SGLOS scheme, the surge guidance law is designed asfollows:

= +u k y Δd e12 2

(12)

where >k 01 , and >Δ 0 is the lookahead distance.In this context, the minimum value of the guided surge velocity can

be obtained

=u k Δdmin 1 (13)

The heading guidance law is governed by

= − − ⎛⎝

⎞⎠

ψ ϕ βy

arctanΔd p d

e

(14)

where

=β v uarctan( / )d d (15)

In addition, the “virtual target” guidance law is chosen as follows:

= + − +u k x U ψ ϕ βcos( )tar e d p d2 (16)

where >k 02 , and = +U u vd d2 2 .

Remark 2. An SGLOS guidance scheme is proposed in (12) and enablesthe surge speed can be flexibly guided by path following error, andthereby enhancing the entire tracking performance. More importantly,the SGLOS scheme can effectively avoid computational complexity andsingularity in previous guidance laws (Fossen et al., 2003; Breivik andFossen, ; Pettersen and Lefeber, 2001; Fossen and Pettersen, 2014;Brhaug et al., 2008; Caharija and Pettersen, 2016; Fossen et al., 2015;Moe et al., 2016).

The key result on the SGLOS scheme is collected as follows.

Theorem 1. Applying the SGLOS guidance scheme (12)–(16) to the pathfollowing system (11), path following errors are globally asymptoticallystable.

Proof. Substituting the heading guidance law in (14) into the fol-lowing formula can be obtained as follows

− = − −

= − −

= −

− = − −

= −

=

+ +

+

+

+ +

−

+

( )

( )

( )

( )

ψ ϕ β

β β

ψ ϕ β

β β

sin( ) sin arctan

sin cos

cos( ) cos arctan

cos sin

d p dyΔ

dΔ

y Δ dy

y Δ

β Δ β y

y Δ

d p dyΔ

dΔ

y Δ dy

y Δ

β Δ β y

y Δ

sin cos

cos sin

e

e

e

e

d d e

e

e

e

e

e

d d e

e

2 2 2 2

2 2

2 2 2 2

2 2 (17)

Together with the SGLOS guidance scheme (12)–(16) and pathfollowing system error dynamics (11), we have

= − +

= − −

+

= − + −+ −

= − −

+

+

−

+

x k x ϕ y

y u ϕ x

u β

k β Δ β y ϕ xk β Δ β y β

y ϕ x

˙ ˙

˙ ˙

tan

(sin cos ) ˙

(cos sin )tan˙

e e p e

eβ Δ β y

y Δ p e

β Δ β y

y Δ

d d e p e

d d ek

β e p e

2

sin cos

cos sin

1

1

cos

d d e

e

d d e

e

d

2 2

2 2

1(18)

Consider the Lyapunov function as follows:

= +V x y12

( )e e12 2

(19)

Differentiating V1 along (18) yields

= − + + ⎡⎣

− − ⎤⎦

= − −

V x k x ϕ y y y ϕ x

y k x

˙ [ ˙ ] ˙e e p e ek

β e p e

kβ e e

1 2 cos

cos2

22

d

d

1

1(20)

Fig. 1. Path following control geometry of a USV.

Fig. 2. The structure design of the FUO-RAPFC scheme.

N. Wang, et al. Ocean Engineering 176 (2019) 57–64

59

Note ≥ >u u 0d dmin and ∈ −β pi pi( /2, /2)d , we further have

≤ − −≤ −

V k y k xkV

˙e e1 12

22

1 (21)

where =k k k2min{ , }1 2 .This concludes the Proof.

4. Controller design

4.1. Fuzzy logic system

A fuzzy logic system (FLS) can be described as follows:

⋯

⋯

R X F X F X F

Y θ Y θ Y θ

: IF is , is , , and is ;

THEN is , is , , and is ,j

j jn n

j

j jm m

j1 1 2 2

1 1 2 2 (22)

where = ⋯ ∈X X X X R[ , , , ]nT n

1 2 and =Y Y Y[ , ,1 2 ⋯ ∈Y R, ]mT m are input

and output variables, respectively, = ⋯ = ⋯F i n j N, 1,2, , ; 1,2, ,ij de-

notes the fuzzy set in the i-th dimension within the j-th fuzzy rule, and= ⋯ = ⋯θ k m j N, 1,2, , ; 1,2, ,k

j denotes the fuzzy singleton.The overall output of the FLS is formulated as follows:

= = ⋯θ ξ XY k m( ), 1,2, ,k kT (23)

where = ⋯θ θ θ θ[ , , , ]k k k kN T1 2 , and =ξ X X Xξ ξ( ) [ ( ), ( ),1 2 ⋯ Xξ, ( )]N

T isfuzzy basis function vector defined by

=∏

∑ ∏= ⋯

=

= =

Xξμ X

μ Xj N( )

( )

( ), 1,2, ,j

in

F i

jN

in

F i

1

1 1

ij

ij (24)

where μ X( )F iij is the membership function of fuzzy set Fi

j and is usuallydefined by Gaussian function.

By the universal approximation capability of the FLS, the unknownuncertainties Du, Dr can be completely described as follows:

= + = +θ ξ θ ξD ε D ε,u uT

u u r rT

r r* * * * (25)

where θu*, θr

* are optimal weight parameters, and εu*, εr

* is the ideal ap-proximation error which is bounded, i.e.,

≤ ≤ε ε ε ε| | ¯ , | | ¯u u r r* * (26)

with an upper bound >ε ε¯ , ¯ 0u r .

4.2. Surge control

Surge control τu is designed as follows:

= − + − − −τ m k u f u v r u D ε u[ ( , , ) ˙ ] ˆ ˆ sgn( )u u e u d u u e11 (27)

where >k 0u , = −u u ue d is the surge tracking error, and D̂u is theoutput of fuzzy unknown observer (FUO) in surge designed as follows:

= θ ξ νD̂ ˆ ( )u uT

u (28)

where ξu is predefined fuzzy basis function, =ν u v r: [ , , ]T is the velocityvector of the USV, and θ̂u and ε̂u are estimates of weight parameters andthe approximation error, respectively, with adaptive laws governed by

= +

= +

θ ξu σ

ε u σ

ˆ̇ ( )

ˆ̇ (| | | |)

uγm e u u

uγm e u

θu

εu

11

11 (29)

where >γ γ, 0θ εu u , and the uncertainty observation error σu is defined asfollows:

= −σ u ρu e u (30)

with auxiliary observation dynamics as follows:

= − + + + −

+ +

ρ k ρ f u v r u

k u

˙ ( , , ) ˙u ρ u uτ

mD

m d

ρ eε σ

m

ˆ

ˆ sgn( )

uu u

uu u

11 11

11 (31)

where >k 0ρu .

Theorem 2. Applying the robust surge control law in (27) and theparameter adaptive laws in (29) with the FUO (28), the guided surgespeed ud in (12) can be globally asymptotically tracked.

Proof. Consider the following Lyapunov function:

⎜ ⎟⎛⎝

⎞⎠

= + + +− −θ θV u γ γ ε σ12

˜ ˜ ˜e θ uT

u ε u u22 1 1 2 2

u u(32)

where = −θ θ θ˜ ˆu u u* and = −ε ε ε˜ ˆu u u

* are the corresponding adaptiveestimation errors.

Differentiating V2 along surge dynamics in (3) yields

= − − +

= ⎡⎣+ + − ⎤⎦

−

− + ⎡⎣⎢

− +

− ⎤⎦⎥

− −

−

−

+

θ θ

θ θ

V u u γ γ ε ε σ σ

u f u v r u γ

γ ε ε σ k σ

˙ ˙ ˜ ˆ̇ ˜ ˆ̇ ˙

( , , ) ˙ ˜ ˆ̇

˜ ˆ̇

e e θ uT

u ε u u u u

e uτ

mD

m d θ uT

u

ε u u u ρ uD

m

D ε σm

21 1

1

1

ˆ ˆ sgn( )

u u

u uu

u uu

u u u

11 11

11

11 (33)

Substituting the robust surge control law in (27) and the parameteradaptive laws in (29) into (33) gives

= ⎡⎣⎢

− + ⎤⎦⎥

−

− + ⎡⎣⎢

− +

+ ⎤⎦⎥

≤ − − + ⎡⎣

+ − ⎤⎦

+ ⎡⎣+ − ⎤⎦

≤ − −

+ − −

−

−

−

−

∗

∗

θ θ

θ ξ θ

V u k u γ

γ ε ε σ k σ

k u k σ u σ γ

ε u σ γ ε

k u k σ

˙ ˜ ˆ̇

˜ ˆ̇

˜ ( ) ˆ̇

˜ (| | | |) ˆ̇

θ ξ

θ ξ

e u eε ε u

m θ uT

u

ε u u u ρ u m

ε ε σm

u e ρ u uT

m e u u θ u

u m e u ε u

u e ρ u

2˜ ˆ sgn( ) 1

1 ˜

ˆ sgn( )

2 2 1 1

1 1

2 2

uT

u u u eu

u uuT

u

u u u

u u

u

u

11

11

11

11

11

(34)

which implies that ue, θ̃u, ε̃u, σu are all bounded. By straightforwardderivation and using Barbalats lemma, we can conclude that ue and σuare globally asymptotically stable.

This concludes the Proof.

4.3. Heading control

Yaw torque τr is designed as follows:

= − + + − − −τ m f u v r k r ψ r D ε r[ ( , , ) ˙ ] ˆ ˆ sgn( )r r r e e d r r e33 (35)

where >k 0r , = −ψ ψ ψe d is the heading tracking error, = −r r re d is theyaw tracking error, and rd is a virtual signal governed by

= − +r k ψ ψ̇d ψ e d (36)

with >k 0ψ , and D̂r is the FUO in yaw dynamics designed as follows:

= θ ξ νD̂ ˆ ( )r rT

r (37)

where ξr is predefined fuzzy basis function, θ̂r and ε̂r are estimates ofweight parameters and approximation error, and are governed by

= +

= +

θ ξr σ

ε r σ

ˆ̇ ( )

ˆ̇ (| | | |)

rγm e r r

rγ

m e r

θr

εr

33

33 (38)

where >γ γ, 0θ εr r , and σr is the uncertainty observation error devised as

N. Wang, et al. Ocean Engineering 176 (2019) 57–64

60

= −σ r ρr e r (39)

with auxiliary observation dynamics as follows:

= − + + + −

+ +

ρ k ρ f u v r r

k r

˙ ( , , ) ˙r ρ r rτ

mD

m d

ρ eε σ

m

ˆ

ˆ sgn( )

rr r

rr r

33 33

33 (40)

where >k 0ρr .The key result on surge and heading tracking is summarized as

follows.

Theorem 3. Consider heading control law in (35) and the parameteradaptive laws in (38) with the FUO (37), the heading tracking error anduncertainty observation error are globally asymptotically stable.

Proof. Consider the following Lyapunov function:

⎜ ⎟⎛⎝

⎞⎠

= + + + +− −θ θV ψ r γ γ ε σ12

˜ ˜ ˜e e θ rT

r ε r r32 2 1 1 2 2

r r(41)

Differentiating V3 along surge dynamics in (3) yields

= + − − +

= + − + ⎡⎣⎢

+

+ − ⎤⎦⎥

− −

+ ⎡⎣

− + ⎤⎦

− −

− −

− −

θ θ

θ θ

V ψ ψ r r γ γ ε ε σ σ

ψ r r ψ r f u v r

r γ γ ε ε

σ k σ

˙ ˙ ˙ ˜ ˆ̇ ˜ ˆ̇ ˙

( ˙ ) ( , , )

˙ ˜ ˆ̇ ˜ ˆ̇

e e e e θ rT

r ε r r r r

e e d d e rτ

m

Dm d θ r

Tr ε r r

r ρ rD D ε σ

m

31 1

1 1

ˆ ˆ sgn( )

r r

r

rr r

rr r r r

33

33

33 (42)

Substituting the robust heading control law in (35) and the para-meter adaptive laws in (38) into (42) gives

= − + ⎡⎣⎢

− + ⎤⎦⎥

− − + ⎡⎣⎢

− +

+ ⎤⎦⎥

≤ − − − + ⎡⎣⎢

+

− ⎤⎦⎥

− + + −

≤ − − −

+ −

− −

−

− −( )

θ θ

ξ

V k ψ r k r

γ γ ε ε σ k σ

k ψ k r k σ θ r σ

γ θ ε r σ γ ε

k ψ k r k σ

˙

˜ ˆ̇ ˜ ˆ̇

˜ ( )

ˆ.

˜ (| | | |) ˆ̂

θ ξ

θ ξ

ψ e e r eε ε rm

θ rT

r ε r r r ρ r m

ε ε σm

ψ e r e ρ r rT

m e r r

θ r r m e r ε r

ψ e r e ρ r

32 ˜ ˆ sgn( )

1 1 ˜

ˆ sgn( )

2 2 2 1

1 1 1

2 2 2

rT

r r r e

r r rrT

r

r r r

r

r r

r

*

33

33

*

33

33

33

(43)

which implies that ψe, re, θ̃r , ε̃r , σr are all bounded. By straightforwardderivation and using Barbalats lemma (Khalil), we can conclude that ψe,re and σr are globally asymptotically stable.

This concludes the Proof.

4.4. Sway dynamics

The sway dynamics can be rearranged into the following form

= − − +m v m ur d v D˙ v22 11 22 (44)

The complex unknowns Dv is bounded in practice, i.e., ≤D D| | ¯v v.Moreover, from Theorems 2 and 3, surge and yaw speeds (u and r) aremade bounded. In this context, we have

≤ − +v dm

v d˙ v22

22 (45)

where d22 is actually positive for an USV (Fossen, 1994), and= + < ∞d m ur D m( ¯ ¯ ¯ )/v v11 22 .Using Bellman-Gronwall comparison principle (Khalil), we have

≤ + −

≤ + ∀ ≤ < ∞

− − − −( )v t v t e e

v t t t

( ) ( ) 1

( ) ,

t t d md

t t

d md

0( ) ( )

0 0

dm v d

m

v

2222 0 22

22

2222 0

2222 (46)

From the above analysis, the sway velocity v is bounded.

5. Simulation studies

In order to verify the proposed FUO-RAPFC scheme, we conductsimulation studies and comparisons on the benchmark USV Cybership I(Ghommam et al., 2007). Inertia parameters of Cybership I are as fol-lows: =m 19kg11 , =m 35.2kg22 , =m 4.2kg33 , =d 4kg/s11 , =d 1kg/s22 ,

=d 10kg/s33 .In order to facilitate simulation settings, parametric uncertainties,

and disturbances are assumed as follows: =m m tΔ 0.1 sin0.711 11 ,=m m tΔ 0.1 cos0.722 22 , =m m tΔ 0.1 sin0.733 33 , =d d tΔ 0.1 cos0.711 11 ,

=d d tΔ 0.1 sin0.722 22 , =d d tΔ 0.1 sin0.733 33 , = +( )τ t2.4sinδπ3u ,

= +( )τ t2.2sinδπ6v , and = +( )τ t0.4sinδ

π3r .

The initial kinematics and dynamics of the USV are as follows:=x y ψ[ , , ] [10,0,0] and =u v r[ , , ] [0,0,0].

The desired path is defined as a curve governed by:

⎧⎨⎩

= +=

x ϖ ϖ ϖy ϖ ϖ

( ) 10 sin(0.1 )( )

p

p (47)

where ϖ t( ) is the time-dependent path variable determined by

=′ + ′

ϖ ux ϖ y ϖ

˙( ) ( )p p

tar2 2

(48)

where utar is the “virtual target” guidance law in (16).In addition, the guidance signals of surge, heading and “target

point” are designed as follows:

⎧

⎨⎪

⎩⎪

= +

= − −

= + − +( )

u k y Δ

ψ ϕ β

u k x U ψ ϕ β

arctan

cos( )

d e

d p dyΔ

tar e d p d

12 2

2

e

(49)

with parameters =k 11 , =k 12 and the lookahead distance =Δ 1.2.The design parameters of the FUO-RAPFC scheme are chosen as

follows: =k 0.25u , =k 0.65ψ , =k 0.25r , =k 3ρu , =k 1.8ρv , =k 8ρr ,= ×γ 2 10θ

3u , = ×γ 5 10θ

4v , = ×γ 3 10θ

3r , =γ 0.03εu , =γ 0.03εv , and

=γ 0.03εr .The actual and reference paths are shown in Fig. 3, in comparison

with the RAPFC scheme without the FUO, the proposed FUO-RAPFCscheme can achieve significantly remarkable performance and muchstronger uncertainty rejection, simultaneously. From Fig. 4, it can be

Fig. 3. Path following performance.

N. Wang, et al. Ocean Engineering 176 (2019) 57–64

61

observed path following errors within the FUO-RAPFC scheme canconverge smoothly to zero while the cross-track error without un-certainty observation cannot converge to zero under complex multipleunknowns. Figs. 5 and 6 show the remarkable path-following perfor-mance of the proposed FUO-RAPFC scheme with multiple uncertainties,whereby the actual signals can track corresponding references. Bycontrast, the RAPFC cannot achieve accurate tracking subject to mul-tiple unknowns. In addition, uncertainty observation errors can con-verge to zero with fast convergence, and thereby contributing to preciseuncertainty observation as shown in Fig. 7. The remarkable perfor-mance of the proposed FUO-RAPFC scheme on exact path followingrelies on the accurate observation on the lumped unknowns via a FUOin Fig. 8. A numerical table of estimated errors is shown in Table 1,among that = −D D D˜ ˆu u u, = −D D D˜ ˆv v v, = −D D D˜ ˆr r r respectivelyrepresent estimation errors in surge, sway and yaw dimensions. Thecontrol input in surge and yaw of the USV are shown in Fig. 9.

In summary, the overall FUO-RAPFC scheme can achieve exact pathfollowing together with accurate uncertainty observation in presence ofmultiple unknowns including the internal parametric perturbations,external environmental disturbances and unmodelled dynamics.

6. Conclusion

In this paper, accurate path following problem of a USV with

Fig. 4. Cross- and along-track errors.

Fig. 5. Surge speed and heading angle tracking.

Fig. 6. Surge and heading tracking errors.

Fig. 7. Uncertainty observation errors.

Fig. 8. Estimates of unknowns.

N. Wang, et al. Ocean Engineering 176 (2019) 57–64

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multiple unknowns has been solved by devising an FUO-RAPFCscheme. Multiple unknowns have been exactly estimated by the pro-posed FUO. In addition, the proposed SGLOS law has achieved si-multaneous surge and heading guidance, and thereby not only enhan-cing path following performance but also avoiding possible singularityand computational complexity of the projection point selection in-volved in traditional LOS methods. Combining backstepping techniquewith the FUO approach, robust adaptive tracking controllers have beendeveloped to render the overall closed-loop tracking error dynamicsglobally asymptotically stable, and thereby contributing to the entireFUO-RAPFC scheme that can achieve accurate path following of a USVwith multiple unknowns.

Acknowledgments

The authors would like to thank the Editor-in-Chief, the AssociateEditor and anonymous referees for their invaluable comments andsuggestions.

Table 1Estimates errors.

Time/s D N˜ /u D N˜ /v D N˜ /r Time/s D N˜ /u D N˜ /v D N˜ /r

1 −0.59084 1.137883 −0.20689 51 0.019496153 −0.01098 0.0058342 −26.4788 0.440689 0.775534 52 0.019448432 −0.01123 0.0058033 5.422698 −4.32615 −4.21744 53 0.019398309 −0.01147 0.0057764 19.16141 −3.54139 0.113111 54 0.019345782 −0.01172 0.0057565 −7.3535 2.726417 0.447984 55 0.019290853 −0.01196 0.0057426 −3.31216 −0.18375 −0.0346 56 0.01923352 −0.0122 0.0057347 3.193548 0.305592 0.125465 57 0.019173783 −0.01245 0.0057348 −0.63752 −0.28335 0.00713 58 0.019111642 −0.01269 0.005749 0.684448 −0.06266 0.030564 59 0.019047098 −0.01293 0.00575410 −0.36464 −0.17738 0.035364 60 0.018980151 −0.01317 0.00577611 0.019411 −0.00114 0.004077 61 0.018910802 −0.01341 0.00580412 0.01946 −0.00139 0.004212 62 0.018839052 −0.01365 0.0058413 0.019507 −0.00163 0.004361 63 0.018764906 −0.01389 0.00588114 0.019552 −0.00187 0.004486 64 0.018688366 −0.01413 0.00592915 0.019595 −0.00212 0.004623 65 0.018609437 −0.01436 0.00598216 0.019635 −0.00236 0.004769 66 0.018528124 −0.0146 0.0060417 0.019672 −0.0026 0.004886 67 0.018444434 −0.01483 0.00610318 0.019708 −0.00285 0.005009 68 0.018358376 −0.01506 0.00616819 0.01974 −0.00309 0.005138 69 0.01826996 −0.01529 0.00623620 0.019771 −0.00334 0.005269 70 0.018179195 −0.01552 0.00630521 0.019799 −0.00358 0.005364 71 0.018086095 −0.01574 0.00637522 0.019824 −0.00383 0.005461 72 0.017990673 −0.01597 0.00644423 0.019847 −0.00407 0.005558 73 0.017895303 −0.0162 0.00653524 0.019868 −0.00432 0.005654 74 0.017797607 −0.01643 0.0066625 0.019886 −0.00457 0.005746 75 0.017698772 −0.01667 0.00679226 0.019902 −0.00481 0.005835 76 0.017597588 −0.0169 0.00695727 0.019915 −0.00506 0.005918 77 0.017494646 −0.01713 0.0071428 0.019926 −0.0053 0.005995 78 0.017389329 −0.01736 0.00735229 0.019934 −0.00555 0.006065 79 0.017281931 −0.01759 0.00758430 0.01994 −0.0058 0.006127 80 0.017172138 −0.01781 0.0078431 0.019943 −0.00604 0.00618 81 0.017060098 −0.01804 0.00811332 0.019944 −0.00629 0.006225 82 0.016945654 −0.01826 0.00840333 0.019942 −0.00654 0.006261 83 0.016828886 −0.01847 0.00870534 0.019938 −0.00678 0.006287 84 0.016709721 −0.01868 0.00901635 0.019932 −0.00703 0.006305 85 0.01658821 −0.01889 0.00933236 0.019923 −0.00728 0.006313 86 0.016464328 −0.01909 0.00964937 0.019911 −0.00752 0.006314 87 0.016338115 −0.01929 0.00996338 0.019897 −0.00777 0.006306 88 0.016209577 −0.01949 0.0102739 0.019881 −0.00802 0.00629 89 0.016078754 −0.01968 0.01056540 0.019862 −0.00826 0.006268 90 0.01594567 −0.01987 0.01084541 0.019841 −0.00851 0.00624 91 0.015810371 −0.02005 0.01110742 0.019817 −0.00876 0.006206 92 0.015672895 −0.02023 0.01134743 0.019791 −0.00901 0.006168 93 0.015533293 −0.02041 0.01156244 0.019763 −0.00925 0.006127 94 0.015391615 −0.02058 0.0117545 0.019732 −0.0095 0.006084 95 0.015247918 −0.02075 0.01190846 0.019699 −0.00975 0.00604 96 0.015102259 −0.02091 0.01203447 0.019663 −0.01 0.005995 97 0.014954701 −0.02108 0.01212748 0.019625 −0.01024 0.005951 98 0.014805306 −0.02124 0.01218649 0.019584 −0.01049 0.005909 99 0.014654141 −0.02139 0.01221150 0.019541 −0.01074 0.00587 100 0.014501269 −0.02155 0.012201

Fig. 9. Control inputs.

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References

Allotta, B., Caiti, A., Costanzi, R., Fanelli, F., Fenucci, D., 2016. Subsea cable tracking byautonomous underwater vehicle with magnetic sensing guidance. Ocean Eng. 113,121–132.

Breivik, M., and Fossen, T.I., “Path following for marine surface vessels. In: Proceedings ofOCEANS’04. MTTS/IEEE TECHNO-OCEAN’04. 2282–2289.

Brhaug, E., Pavlov, A., Pettersen, K.Y., 2008. Integral LOS control for path following ofunderactuated marine surface vessels in the presence of constant ocean currents. In:Proceedings of IEEE Conference on Decision and Control, pp. 4984–4991.

Caharija, W., Pettersen, K.Y., 2016. Integral line-of-Sight guidance and control of un-deractuated marine vehicles: theory, simulations, and experiments. IEEE Trans.Control Syst. Technol. 24 (5), 1623–1642.

Fossen, T.I., Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley &Sons, Ltd.

Fossen, T.I., 1994. Guidance and Control of Ocean Vehicle. Wiley, New York.Fossen, T.I., Pettersen, K.Y., 2014. On uniform semiglobal exponential stability (USGES)

of proportional line-of-sight guidance laws. Automatica 50 (11), 2912–2917.Fossen, T.I., Breivik, M., Skjetne, R., 2003. Line-of-sight path following of underactuated

marine craft. In: Proceedings of the 6th IFAC Conference on Manoeuvring andControl of Marine Craft. MCMC), pp. 211–216.

Fossen, T.I., Pettersen, K.Y., Galeazzi, R., 2015. Line-of-sight path following for dubinspaths with adaptive sideslip compensation of drift forces. IEEE Trans. Control Syst.Technol. 23 (2), 820–827.

Ghommam, J., Mnif, F., Benali, A., Poisson, G., 2007. Observer design for euler Lagrangesystems: application to path following control of an underactuated surface vessel. In:Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems.IROS, pp. 2883–2888.

Khalil, H., Nonlinear Systems, second ed. Prentice-Hall Press, Upper Saddle River, NJ,USA.

Li, J.H., Lee, P.M., Jun, B.H., 2008. Point-to-point navigation of underactuated ships.Automatica 44 (12), 3201–3205.

Li, H., Bai, L., Wang, L., Zhou, Q., Wang, H., 2017. Adaptive neural control of uncertainnonstrict-feedback stochastic nonlinear systems with output constraint and unknowndead zone. IEEE Trans. Syst. Man. Cybern. Syst. 47 (8), 2048–2059.

Liu, S., Li, Y., Wang, N., 2017. Nonlinear disturbance observer-based backstepping finite-time sliding mode tracking control of underwater vehicles with system uncertaintiesand external disturbances. Nonlinear Dynam. 88 (1), 465–476.

Mai, T.V., Choi, H.S., Kang, J., Ji, D.H., Jeong, S.K., 2017. A study on hovering motion ofthe underwater vehicle with umbilical cable. Ocean Eng. 135, 137–157.

Miao, J., Wang, S., Zhao, Z., Li, Y., Tomovic, M.M., 2017. Spatial curvilinear path fol-lowing control of underactuated AUV with multiple uncertainties. ISA Trans. 67,107–130.

Moe, S., Pettersen, K.Y., Fossen, T.I., 2016. Line-of-sight curved path following for un-deractuated USVs and AUVs in the horizontal plane under the influence of oceancurrents. In: Proceeding of 24th Mediterranean Conference on Control andAutomation, pp. 38–45.

Pettersen, K.Y., Lefeber, E., 2001. Waypoint tracking control of ships. In: Proceedings ofIEEE Conference on Decision and Control, pp. 940–945.

Wang, N., Er, M.J., 2015. Self-constructing adaptive robust fuzzy neural tracking controlof surface vehicles with uncertainties and unknown disturbances. IEEE Trans. ControlSyst. Technol. 23 (3), 991–1002.

Wang, N., Sun, J.C., Er, M.J., Liu, Y.C., 2015a. A novel extreme learning control frame-work of unmanned surface vehicles. IEEE Trans. Cybern. 46 (5), 1106–1117.

Wang, N., Qian, C.J., Sun, J.C., Liu, Y.C., 2016a. Adaptive robust finite-time trajectorytracking control of fully actuated marine surface vehicles. IEEE Trans. Control Syst.Technol. 24 (4), 1454–1462.

Wang, N., Qian, C.J., Sun, Z.Y., 2017a. Global asymptotic output tracking of nonlinearsecond-order systems with power integrators. Automatica 80, 156–161.

Wang, N., Sun, J.C., Han, M., Zheng, Z., Er, M.J., 2017b. Adaptive approximation-basedregulation control for a class of uncertain nonlinear systems without feedback line-arizability. IEEE Trans. Neural Netw. Learn. Syst. 29 (8), 3747–3760.

Wang, N., Lv, S., Zhang, W., Liu, Z., Er, M.J., 2017d. Finite-time observer based accuratetracking control of a marine vehicle with complex unknowns. Ocean Eng. 145,406–415.

Wang, L., Basin, M., Li, H., Lu, R., 2017e. Observer-based composite adaptive fuzzycontrol for nonstrict-feedback systems with actuator failures. IEEE Trans. Fuzzy Syst.26 (4), 2336–2347.

Wang, N., Su, S.F., Yin, J.C., Zheng, Z.J., Er, M.J., 2017f. Global asymptotic model-freetrajectory-independent tracking control of an uncertain marine vehicle: an adaptiveuniverse-based fuzzy control approach. IEEE Trans. Fuzzy Syst. 26 (3), 1613–1625.

Wang, N., Xe, G., Pan, X., Su, S.-F., 2018a. Full-state regulation control of asymmetricunderactuated surface vehicles. IEEE Trans. Ind. Electron. https://doi.org/10.1109/TIE.2018.2890500.

Wang, N., Su, S.-F., Pan, X., Yu, X., Xie, G., 2018b. Yaw-guided trajectory tracking controlof an asymmetric underactuated surface vehicle. Ind. Inf. IEEE Trans. https://doi.org/10.1109/TII.2018.2877046.

Wang, N., Su, S.-F., Han, M., Chen, W.-H., 2018d. Backpropagating constraints-basedtrajectory tracking control of a quadrotor with constrained actuator dynamics andcomplex unknowns. IEEE Trans. Syst. Man Cybern. A Syst. https://doi.org/10.1109/TSMC.2018.2834515.

Wang, N., Sun, J.C., Er, M.J., 2018e. Tracking-error-based universal adaptive fuzzycontrol for output tracking of nonlinear systems with completely unknown dynamics.IEEE Trans. Fuzzy Syst. 26 (2), 869–883.

Wang, N., Deng, Q., Pan, X., Xie, G., 2019. Hybrid finite-time trajectory tracking controlof a quadrotor. ISA (Instrum. Soc. Am.) Trans. https://doi.org/10.1016/j.isatra.2018.12.042.

Wynn, R.B., Huvenne, V.A., 2014. Autonomous underwater vehicles: their past, presentand future contributions to the advancement of marine geoscience. Mar. Geol. 352(2), 451–468.

Xiang, X., Yu, C., Niu, Z., Zhang, Q., 2016. A new AUV navigation system exploitingunscented Kalman filter. Sensors 16 (8), 1335.

Xu, Y., Xiao, K., 2007. Technology development of autonomous ocean vehicle. ActaAutom. Sin. 33 (5), 518–521.

Yi, B., Qiao, L., Zhang, W., 2016. Two-time scale path following of underactuated marinesurface vessels: design and stability analysis using singular perturbation methods.Ocean Eng. 124, 287–297.

Zhao, X., Liu, Y., Han, M., Wu, D., Li, D., 2016. Improving the performance of an AUVhovering system by introducing low-cost flow rate control into water hydraulicvariable ballast system. Ocean Eng. 125, 155–169.

Zhou, Q., Li, H., Wang, L., Lu, R., 2017a. Prescribed performance observer-based adaptivefuzzy control for nonstrict-feedback stochastic nonlinear systems. IEEE Trans. Syst.Man. Cybern. Syst. 48 (10), 1747–1758.

N. Wang, et al. Ocean Engineering 176 (2019) 57–64

64