barrier lyapunov function-based adaptive control of an

Research ArticleBarrier Lyapunov Function-Based Adaptive Control of anUncertain Hovercraft with Position and Velocity Constraints

Mingyu Fu Taiqi Wang and ChenglongWang

College of Automation Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Taiqi Wang wangtaiqihrbeueducn

Received 20 November 2018 Accepted 27 January 2019 Published 12 February 2019

Academic Editor Sergey Dashkovskiy

Copyright copy 2019 Mingyu Fu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers the problem of constrained path following control for an underactuated hovercraft subject to parametricuncertainties and external disturbances A four-degree-of-freedom hovercraft model with unknown curve-fitted coefficients isfirst rewritten into a parameterized form By introducing a barrier Lyapunov function into the line-of-sight guidance the specifictransient tracking performance in terms of position error is guaranteed A novel constrained yaw rate controller is proposed toensure time-varying yaw rate constraint satisfaction inwhich the yaw rate barrier is required to varywith the speed of the hovercraftMoreover a command filter is incorporated into the control design to generate the desired virtual controls and its time derivativesTheoretical analyses show that under the proposed controller the position tracking error constraints and the yaw rate constraintcan be strictly guaranteed Finally numerical simulations illustrate the effectiveness and advantages of the proposed control scheme

1 Introduction

As a high-performance amphibious marine craft a hover-craft utilizes a flexible skirt system around its peripherysuch that the hull is totally supported by a pressurized aircushion The hovercraft has attracted increasing attentionin both military and civil fields because of its superiorhigh speed and amphibious characteristics [1] Due to itscomplex wave-making resistance and skirt drag caused bythe cushion system the dynamics of a hovercraft are veryuncertain nonlinear and coupled [2] Moreover a hovercraftis underactuated because the actuators are equipped forsurge and yaw motion only [3] In addition a hovercraft isessentially hovering over the water surface which causes lesswater friction drag than conventional displacement shipsTherefore a hovercraft can slip considerably and undergogreat heeling during fast turning requiring the yaw rate ofa hovercraft to be regulated within specific safe ranges duringmaneuvering These requirements make controller design ofa hovercraft a challenging task

Path following which requires a hovercraft to follow ageometric path that is time independent is one of the typicalcontrol scenarios for a marine surface vessel (MSV) In [4]a global path following controller was designed for MSVs

based on a cascaded approach and the stability was proved byusing the linear time-varying theory Reference [5] utilized abackstepping technique to develop a nonlinear path followingcontroller for MSVs in which the control design was basedon feedback dominance instead of feedback linearizationReference [6 7] presented a model predictive control schemewith line-of-sight (LOS) guidance to improve the path follow-ing performance Reference [8] proposed an adaptive pathfollowing controller to estimate the ocean currents whilea new integral LOS guidance law was obtained based onadaptive control Neural network (NN) control is usuallyused to deal with the uncertain nonlinear system [9] whichhas also been widely introduced into path following controlto cope with the uncertain dynamics of surface vessels In[10 11] the NN control approach together with the integralLOS guidancewas proposed for underactuated surface vesselswith parameter uncertainties Reference [12] developed asaturated path following controller for surface vessels and theuncertainties and disturbances were approximated by usingNN

One of the greatest challenges for hovercraft controlis the inexact dynamics which are due to the intricateinteractions between the cushion system and the watersurface these interactions lead to an unclear hydrodynamic

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1940784 16 pageshttpsdoiorg10115520191940784

2 Mathematical Problems in Engineering

structure and parametric uncertainties for control design Anexceedingly simplified hovercraft model was derived in [13ndash15] in which the hydrodynamic damping coefficients wereassumed to be zero The same hovercraft model was adoptedin [16 17] However because the hydrodynamic force andmoment are not included in this model the proposedmodel-based controllers might not achieve the desired controlobjective in practical applications Furthermore a curve-fitted hovercraft model was obtained in [1] by replacing thecomplex hydrodynamic and aerodynamic force and momentfunctionswith curve-fitted approximations by neglecting theheave and pitch motion a four-degree-of-freedom (DOF)control-oriented hovercraft model was obtained for controldesign However the effects of parametric uncertaintiesand external disturbances were not considered in [1] Theuncertainties and disturbances always exist in a practicalcontrol system and some works have investigated the feed-back control schemes for different form uncertain nonlinearsystems such as triangular form systems [18 19] and strict-feedback systems [20] In addition the adaptive controlmethod is regarded as a powerful method to cope withsuch system uncertainties [21] For example [22] presenteda backstepping-based adaptive control scheme to estimatethe unknown parameters and the tuning-function basedapproach was proposed to avoid the overparametrization inbackstepping design The adaptive backstepping techniquehas been widely applied in path following control of vessels[23ndash25] To improve the tracking performance of a hovercraftunder these system uncertainties the above 4-DOF hover-craft dynamics [1] are rewritten into a parameterized formin this paper and the adaptive control method is adopted toestimate the uncertain parameters and external disturbances

LOS guidance which has been widely explored forsurface vessels due to its simplicity and small computationalburden is an effective guidance algorithm for path followingcontrol In [26 27] it was shown that the equilibrium pointof the proportional LOS guidance law is uniformly andexponentially semi-globally stable To compensate for thevehiclersquos constant sideslip due to environmental disturbancesan integral LOS (ILOS) approach was proposed in [28 29]by adding an additive integral action into the conventionalLOS guidance In [10 30] an adaptive LOS (ALOS)-basedcontroller was developed with a parameter adaption law toeliminate the effects of system uncertainties In [31] theauthors discussed the drawbacks of the ILOS and ALOSA modified extended state observer- (ESO-) based LOSguidance approachwas proposed to identify the time-varyingsideslip angle Reference [32] presented a compound LOSguidance by combining the time delay control and ESOtechnique to estimate the time-varying ocean currents Notethat all the aforementioned LOS guidance algorithms onlyregulate the steady state of position tracking errors to bezero while the transient tracking performance cannot beguaranteed However it is important to ensure the prescribedtransient tracking performance of a hovercraft in practiceMoreover the yaw rate of a hovercraft should be limited tobelow the stability boundary for safe navigation [33] If themaximum yaw rate of a hovercraft exceeds the correspondingstability boundary for a period of time the hovercraft will

become unstable especially at high speeds which can evenlead to a capsizing hazard As shown in [33] the stabilityboundary of the yaw rate varies with the surge speed ofa hovercraft Thus the yaw rate constraint should also betime varying because the desired speed command for ahovercraft can change in practice To the authorrsquos knowledgethe problem of tracking error-constrained path followingcontrol for an uncertain hovercraft with a time-varying yawrate constraint has rarely been considered

To guarantee the state or output constraints of thesystem some control methods have been included in modelpredictive control [34] nonovershooting control [35] andprescribed performance control [36] However in thesemethods the state constraints remain difficult to guaranteein practical applications More recently a barrier Lyapunovfunction (BLF) has been proposed for nonlinear systems toensure the state and output constraints [37 38] which hasbeen applied to strict-feedback systems with output con-straints [39] pure-feedback systems [40] switched systems[41] and practical applications for hypersonic flight vehicles[42] or missile guidance [43] In addition to the conventionallogarithmic function form a modified tan-type BLF wasproposed in [44] which is a general method for systems withstate constraints because it works even if the state constraintsare removed In [45] the authors utilized the tan-type BLFto design an error-constrained LOS path following controllerfor a 3-DOF conventional surface vessel To break the staticconstraint limitations in the abovementioned works a time-varying output constraint was handled in [46] by using atime-varying BLF which allowed the output constraints tobe both time varying and asymmetric Moreover to facilitatethe backstepping design for the attitude subsystem of ahovercraft a command filter was introduced to avoid theanalytical computation of the virtual control laws [47]

Motivated by the above considerations a BLF-basedadaptive path following control scheme is developed for a 4-DOF underactuated hovercraft subject to parametric uncer-tainties and external disturbancesThemain contributions ofthis paper can be summarized as follows(1) A novel position-constrained LOS guidance algo-rithm is proposed by introducing BLF into the classical LOSguidance design procedure The proposed LOS guidance canguarantee the prescribed tracking performance in terms ofposition tracking errors(2) By virtue of the time-varying BLF an attitude con-troller is proposed to ensure the yaw rate time-varyingconstraints of a hovercraft The yaw rate of a hovercraftwill not exceed the stability boundary which has practicalsignificance for the safe navigation of a hovercraft(3) A command filter is integrated into the controldesign to generate the amplitude-constrained virtual controllaws which avoid the analytic computation of the virtualcontrol law derivative In addition a novel auxiliary systemis developed to compensate for the filtering error

The remainder of this paper is organized as follows Theproblem formulation is introduced in Section 2 Section 3 isdevoted to the LOS guidance altitude subsystem and velocitysubsystem control design Numerical simulation examplesare presented in Section 4 Section 5 concludes the paper

Mathematical Problems in Engineering 3

xE

OE yE

zE

(sway)

yb

u((surge)p(roll)

xbr(yaw)

zb

Ob

Figure 1 The hovercraft model in the body-fixed frame

2 Problem Formulation and Preliminaries

21 Preliminaries

Notation Throughout this paper (sdot)T denotes the transposeof a matrix (sdot) | sdot | represents the absolute value of a scalar sdot represents the Euclidean norm of a vector

and(sdot) denotes theestimate of (sdot) the estimation error is defined as (sdot) = (sdot)minus and(sdot)and R+ denotes the set of nonnegative real numbers

Lemma 1 (see [44]) For any 120575 gt 0 and 120578 isin R the followinginequality always holds

0 le 10038161003816100381610038161205781003816100381610038161003816 minus 120578 tanh(120578120575) le 02785120575 (1)

Lemma 2 (see [37]) For any positive constant 119896119887 positiveinteger 119901 and 119911 isin R satisfying |119911| lt 119896119887 there exists

log11989621199011198871198962119901119887 minus 1199112119901 lt 11991121199011198962119901119887 minus 1199112119901 (2)

Definition 3 (see [39]) A barrier Lyapunov function is ascalar function 119881(119909) defined with respect to the system = 119891(119909) on an open region 119863 containing the originIt is continuous is positive definite has continuous first-order partial derivatives at every point of119863 has the property119881(119909) 997888rarr infin as 119909 approaches the boundary of119863 and satisfies119881(119909) le 119887 forall119905 ge 0 along the solution of = 119891(119909) for 119909(0) isin 119863and some positive constant 11988722 Dynamic Model of a Hovercraft The typical configura-tion of the hovercraft is shown in Figure 1 in which two setsof ducted air propellers are mounted at the stern and a pairof rudders is mounted behind the duct flaps to create turningmoments

For motion control of the hovercraft we first definethe earth-fixed frame 119874119864119909119864119910119864119911119864 and the body-fixed frame119874b119909b119910b119911b The origin of the body-fixed frame is located atthe center of gravity (CG) By neglecting the pitch and heave

motion and making some basic assumptions the 4-DOFcurve-fitted hovercraft dynamics are derived in [1] and areformulated as follows

= V119903 + (119865119909119886 + 119865119909ℎ + 119865119909119899 + 119865119909119879)119898V = minus119906119903 + (119865119910119886 + 119865119910ℎ + 119865119910119888 + 119865119910119899)119898119903 = (119872119911119886 +119872119911ℎ +119872119911119888 +119872119911119899 +119872119911120575)119868119911

= (119872119909119886 +119872119909ℎ +119872119909119888 +119872119909119899 +119872119909119866)119868119909

(3)

where 119898 is the mass of the hovercraft 119868119909 and 119868119911 are themoments of inertia of the hovercraft about the 119909119887 axis and119911b axis respectively 119906 and V are the surge and sway linearvelocities respectively and 119901 and 119903 represent the roll and yawangular velocities respectively The right side of (3) consistsof various forces and moments acting on the hovercraftin which the suffix 119886 represents the aerodynamic force ℎdenotes the hydrodynamic force 119888 denotes the cushion force119899 denotes the air momentum force 119879 denotes the thrust120575 denotes the rudder moment and 119866 denotes the restoringmoment during heeling of the hovercraft The curve-fittedaerodynamic forces and moments in (3) are written as

119865119909119886 = 051205881198861198812119886 119878119909119886 (1198621199091198860 + 119862120575119903119909119886120575119903 + 119862120573119909119886120573 + 119862 120573119909119886

120573)119865119910119886 = 051205881198861198812119886 119878119910119886 (1198621199101198860 + 119862120575119903119910119886120575119903 + 119862120573119910119886120573 + 119862 120573

119910119886120573)

119872119911119886 = 051205881198861198812119886 119878119911119886119897119886 (1198621198981199111198860 + 119862120575119903119898119911119886120575119903 + 119862120573119898119911119886120573 + 119862 120573119898119911119886

120573)119872119909119886 = 119865119910119886119911119886

(4)

In addition the curve-fitted hydrodynamic forces andmoments in (3) are written as

119865119909ℎ = 051205881199081198802119878ℎ (119862119909ℎ0 + 119862120601119909ℎ120601 + 119862120573119909ℎ120573)119865119910ℎ = 051205881199081198802119878ℎ (119862119910ℎ0 + 119862120601119910ℎ120601 + 119862120573119910ℎ120573)119872119911ℎ = 051205881199081198802119878ℎ119897119888 (119862120601119898119911ℎ120601 + 119862120573119898119911ℎ120573)119872119909ℎ = 119865119910ℎ119911119892

(5)

where119881119886 denotes the relative wind speed 119878119909119886 119878119910119886 and 119878119911119886 arethe projection areas of the hovercraft along the three axesTheaverage length of the hovercraft is denoted by 119897119886 the verticaldistance from the center of the lateral area is denoted by 119911119886the velocity of the hovercraft is denoted by 119880 = radic1199062 + V2the lateral area of the skirts in contact with the water surfaceis denoted by 119878ℎ the cushion length is denoted by 119897119888 and theheight of CG is defined as 119911119892 120575119903120601 and120573 represent the rudderangle heeling angle and sideslip angle of the hovercraftrespectively 119862

lowast represents the curve-fitted coefficient Theexpressions of other component forces and moments can befound in [1] (Chapter 6)


In this paper we assume that 119878119909119886 119878119910119886 119878119911119886 119878ℎ and 119862lowast in

(4) and (5) are unknown constants Thus (4) and (5) can beparameterized into the following form

119865119909119886119898 = 120599T11990611205931199061119865119910119886119898 = 120599TV1120593V1119872119911119886119868119911 = 120599T11990311205931199031119872119909119886119868119909 = 120599T11990111205931199011119865119909ℎ119898 = 120599T11990621205931199062119865119910ℎ119898 = 120599TV2120593V2119872119911ℎ119868119911 = 120599T11990321205931199032119872119909ℎ119868119909 = 120599T11990121205931199012

(6)

By substituting (6) and other component forces andmomentsinto (3) a 4-DOF parameterized hovercraft model can begiven as

= V119903 + 119891119906 + 120599T11990611205931199061 + 120599T11990621205931199062 + 120591119906 + 119889119906V = minus119906119903 + 119891V + 120599TV1120593V1 + 120599TV2120593V2 + 119889V119903 = 119891119903 + 120599T11990311205931199031 + 120599T11990321205931199032 + 120591119903 + 119889119903 = 119891119901 + 120599T11990111205931199011 + 120599T11990121205931199012 + 119889119901

(7)

where the control inputs in (7) are 120591119906 = 119865119909119879119898 and120591119903 = 119872119911120575119868119911 The dynamics 119891119906 119891V 119891119901 and 119891119903 are theintegration of other known component forces and momentsin (3) which are defined as 119891119906 = 119865119909119899119898 119891V = (119865119910119888 +119865119910119899)119898 119891119903 = (119872119911119888 + 119872119911119899)119868119911 and 119891119901 = (119872119909119888 + 119872119909119899 +119872119909119866)119868119909 In addition 119889119906 119889V 119889119903 and 119889119901 in (7) denote thebounded disturbances including the unmodeled dynamicsand external disturbances induced by wind and waves SeeAppendix B for the detailed expressions of the parameters in(7)

The desired parameterized path is defined by 119883119901 =(119909119901(120596) 119910119901(120596)) where 120596 isin R denotes the path variable Apath-fixed reference frame is defined with its origin at 119883119901The path-tangential angle is calculated by

120574119901 (120596) = atan2 (1199101015840119901 (120596) 1199091015840119901 (120596)) (8)

where 1199091015840119901(120596) = 120597119909119901120597120596 and 1199101015840119901(120596) = 120597119910119901120597120596 For a hovercraftlocated at 119883 = (119909 119910) the geometry of LOS guidance isillustrated in Figure 2

North

East Xp

pxp

s

yp

e

u

U

X

Figure 2 The LOS guidance geometry

The along-track error 120585119904(119905) and the cross-track error 120585119890(119905)in the path-fixed frame are written as

[120585119904120585119890] = [cos 120574119901 minus sin 120574119901sin 120574119901 cos 120574119901 ]

T [119909 minus 119909119901119910 minus 119910119901] (9)

According to the kinematic equations of surface vessels givenin [48] the 4-DOF kinematics of the hovercraft is expressed as

[[[[[[

120601]]]]]]= [[[[[[

cos120595 minus sin120595 cos120601 0 0sin120595 cos120595 cos120601 0 00 0 cos120601 00 0 0 1

]]]]]][[[[[[

119906V119903119901]]]]]]

(10)

where 119909 and 119910 are the coordinates of the hovercraftrsquos CG inthe earth-fixed frame120595 is the heading angle of the hovercraftand 120601 is the heeling angle of the hovercraft

The time derivative of the tracking errors in (9) is written as

[ 120585119904120585119890] = 120574119901 [[[[[minus sin 120574119901 (119909 minus 119909119901) + cos 120574119901 (119910 minus 119910119901)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120585119890minus cos 120574119901 (119909 minus 119909119901) minus sin 120574119901 (119910 minus 119910119901)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟minus120585119904

]]]]]+ [[

cos 120574119901 ( minus 119901) + sin 120574119901 ( 119910 minus 119910119901)minus sin 120574119901 ( minus 119901) + cos 120574119901 ( 119910 minus 119910119901)]](11)

Substituting (8) and (10) into (11) yields

[ 120585119904120585119890] = [[119906 cos (120595 minus 120574119901) minus V sin (120595 minus 120574119901) minus radic11990910158402119901 + 11991010158402119901119906 sin (120595 minus 120574119901) + V cos (120595 minus 120574119901) ]]+ [ 120574119901120585119890minus 120574119901120585119904]

(12)

Setting 120573 = atan2(V 119906) (12) can finally be written as

[ 120585119904120585119890] = [[119880 cos (120595 minus 120574119901 + 120573) minus radic11990910158402119901 + 11991010158402119901 + 120574119901120585119890119880 sin (120595 minus 120574119901 + 120573) minus 120574119901120585119904 ]] (13)

where V = V cos120601 is the projection of the lateral velocity ontothe horizontal plane 119880 = radic1199062 + V2 is the hovercraft speed If


the roll motion is not considered we obtain cos120601 = 1 Theobtained equation (13) is similar to the previous results [8 31]for 3-DOF surface vessels

Suppose that 119896119904 119896119890 and 119896119906119890

are positive constant con-straints and 119896119903(119905) is the symmetric time-varying yaw rateconstraint where the time derivatives of 119896119904 119896119890 119896119906

119890

and 119896119903(119905)are bounded The control objective of this study is to designa constrained path following controller for a hovercraftin the presence of parametric uncertainties and externaldisturbances that can guarantee the prescribed trackingperformance |120585119904| lt 119896119904 and |120585119890| lt 119896119890 The yaw rate satisfiesthe constraint |119903(119905)| lt 119896119903(119905) and the surge velocity satisfiesthe constraint |119906119890| lt 119896119906

119890

for all 119905 gt 0 where 119906119890 = 119906 minus 119906119889Remark 4 In the previous path following literature [8 10 2931] the proposed LOS guidance algorithms only regulate thesteady-state tracking errors to be zero or within the neighbor-hood of zero while the transient tracking performances arenot considered Reference [45] also proposed a LOS guidanceto limit the position tracking errors However only theoutput constraints of the vessel were guaranteed the yaw rateand velocity constraints were not guaranteed In this paperboth the state and the output constraints of vessel systemcan be strictly guaranteed by the proposed control schemeIn addition the yaw rate of a hovercraft is of particularconcern to sailors during maneuver Because the hovercraftusually hovers over the water with a high forward velocitya large yaw rate or an excessive rudder angle can easilyinfluence the navigation safety of a hovercraft as compared toa conventional displacement ship Reference [33] shows thatthe stability boundary of the yaw rate varies with the surgespeed of the hovercraft which leads to the conversion of ourdesign task into a time-varying state constrained problem

To facilitate the control design we make the followingassumptions

Assumption 5 The disturbances 119889119906(119905) 119889V(119905) 119889119903(119905) and 119889119901(119905)are boundedThere exist unknown positive constants119863119906119863V119863119903 and 119863119901 such that |119889119906| le 119863119906 |119889V| le 119863V |119889119903| le 119863119903 and|119889119901| le 119863119901 hold for all 119905 ge 0Assumption 6 The initial tracking errors 120585119904(0) 120585119890(0) and119906119890(0) satisfy the prescribed constraints |120585119904(0)| lt 119896119904 |120585119890(0)| lt119896119890 and |119906119890(0)| lt 119896119906

119890

and the yaw rate tracking error satisfies|119903119890(0)| lt 119896119903119890

(0)Assumption 7 For any 119896119904 gt 0 and 119896119890 gt 0 the first andsecond time derivatives of the desired path (119909119901 119910119901) are bothbounded

Assumption 8 Due to underactuated configuration of thehovercraft a constraint on sway dynamics without controlinput is required that the sway velocity of the hovercraft ispassive bounded

3 Control System Design

31 Position-Constrained LOS Guidance In this subsectionthe barrier Lyapunov function will be introduced into the

LOS guidance to guarantee the position constraints TheLyapunov function is defined as follows

1198811 = 12 log 11989621199041198962119904 minus 1205852119904 + 12 log 11989621198901198962119890 minus 1205852119890 (14)

The time derivative of 1198811 is calculated as

1 = 120599119904 120585119904 + 120599119890 120585119890 (15)

where 120599119904 = 120585119904(1198962119904 minus 1205852119904 ) and 120599119890 = 120585119890(1198962119890 minus 1205852119890 )Substituting (13) into (15) we obtain

1 = 120599119904 (119880 cos (120595 minus 120574119901 + 120573) minus radic11990910158402119901 + 11991010158402119901 + 120574119901120585119890)+ 120599119890 (119880 sin (120595 minus 120574119901 + 120573) minus 120574119901120585119904)

= 120599119904 (119880 cos (120595 minus 120574119901 + 120573) minus radic11990910158402119901 + 11991010158402119901 )+ 120599119904 120574119901120585119890 (1 minus 1198962119904 minus 12058521199041198962119890 minus 1205852119890 ) + 120599119890119880 sin (120595 minus 120574119901 + 120573)

(16)

Taking the time derivative of 120574119901 yields120574119901 = 119910101584010158401199011199091015840119901 minus 11990910158401015840119901119910101584011990111990910158401199012 + 11991010158401199012 (17)

Invoking (17) in (16) we have

1 = 120599119904 (119880 cos (120595 minus 120574119901 + 120573) minus 120590)+ 120599119890119880 sin (120595 minus 120574119901 + 120573) (18)

with

120590 = radic11990910158402119901 + 11991010158402119901 minus 120585119890119910101584010158401199011199091015840119901 minus 11990910158401015840119901119910101584011990111990910158401199012 + 11991010158401199012 (1 minus 1198962119904 minus 12058521199041198962119890 minus 1205852119890 ) (19)

Defining the tracking error as

120595119890 = 120595 minus 120595119889 (20)

and substituting (20) into (18) yield

1 = 120599119904 (119880 cos (120595 minus 120574119901 + 120573) minus 120590)+ 120599119890 ((119906119889 sin (120595119889 minus 120574119901) + V cos (120595119889 minus 120574119901)) cos120595119890+ 119906119890 sin (120595 minus 120574119901)+ (119906119889 cos (120595119889 minus 120574119901) minus V sin (120595119889 minus 120574119901)) sin120595119890)= 120599119904 (119880 cos (120595 minus 120574119901 + 120573) minus 120590)+ 120599119890 (119880119889 sin (120595119889 minus 120574119901 + 120573119889) cos120595119890 + 119906119890 sin (120595 minus 120574119901)+ 119880119889 cos (120595119889 minus 120574119901 + 120573119889) sin120595119890)

(21)

with 119880119889 = radic1199062119889 + V2 and 120573119889 = atan2(V 119906119889)


From (21) we can obtain the update law for 120596 = 119880 cos (120595 minus 120574119901 + 120573) + 1198961120585119904120590 (22)

Note that 120590 is the denominator of (22) to avoid 120590 = 0 we cantake 120590 = 1205900 sgn(120590)when |120590| lt 1205900 where 1205900 is a small positiveconstant

The desired heading angle of the hovercraft is chosen tobe 120595119889 = 120574119901 minus 120573119889 + atan2 (minus120585119890 1198962) (23)

where 1198961 1198962 gt 0 are positive design constants and 1198962 is thelook-ahead distance

By invoking (22) and (23) (21) becomes

1 = minus119896112058521199041198962119904 minus 1205852119904 + 120599119890(minus cos120595119890119880119889120585119890 + sin1205951198901198801198891198962radic11989622 + 1205852119890+ 119906119890 sin (120595 minus 120574119901))

= minus 119896112058521199041198962119904 minus 1205852119904 minus 119880119889radic11989622 + 120585211989012058521198901198962119890 minus 1205852119890 + 120599119890120595119890120594 + 120599119890119906119890

sdot sin (120595 minus 120574119901) le minus12058911198811 + 120599119890120595119890120594 + 120599119890119906119890 sin (120595 minus 120574119901)

(24)

where

120594 = minus 119880119889120585119890radic11989622 + 1205852119890(cos120595119890 minus 1)120595119890 + 1198962119880119889radic11989622 + 1205852119890

sin120595119890120595119890 (25)

with 1205891 = min1198961 119880119889radic11989622 + 1205852119890 Remark 9 In (25) we have |120585119890|radic11989622 + 1205852119890 le 1 and1198962radic11989622 + 1205852119890 le 1 By using LrsquoHopitalrsquos rule we obtainlim120595

119890997888rarr0((cos120595119890 minus 1)120595119890) = 0 and lim120595

119890997888rarr0(sin120595119890120595119890) = 1

Thus singularity will not occur in (25)

32 Attitude Control In this subsection the attitude kine-matic controller for the hovercraftwill be designed to stabilizethe yaw angle tracking error 120595119890 Subsequently the yaw ratecontroller will be developed via BLF-based adaptive controlwhile the time-varying constraint on the yaw rate |119903| lt 119896119903(119905)will be strictly guaranteed

Suppose that the virtual yaw rate 120572119903 satisfies10038161003816100381610038161205721199031003816100381610038161003816 lt 119896120572119903 (119905) lt 119896119903 (119905) (26)

where 119896120572119903

(119905) R+ 997888rarr R+and 119896119903(119905) R+ 997888rarr R+The yaw ratetracking error is defined as 119903119890 = 119903minus120572119903 If a yaw rate controllerexists that can ensure the inequality10038161003816100381610038161199031198901003816100381610038161003816 lt 119896119903119890 (119905) le 119896119903 (119905) minus 119896120572119903 (119905) (27)

where 119896119903119890

(119905) R+ 997888rarr R+ and then the time-varyingconstraint 119896119903(119905) on yaw rate 119903(119905) can be guaranteed

Magnitude limiter

rckr

(t)

-kr(t)

+ +

- -

2n

2nn2nn

1

s

1

s

r

Figure 3 Configuration of the command filter

Remark 10 If the designed yaw rate controller canmake |119903119890| lt119896119903119890

(119905) then we have |119903 minus 120572119903| lt 119896119903(119905) minus 119896120572119903

(119905) and |119903| minus |120572119903| lt119896119903(119905)minus119896120572119903

(119905) Using (26) we obtain |119903| lt 119896119903(119905)minus(119896120572119903

(119905)minus120572119903) lt119896119903(119905)However one problem is how to ensure the inequality|120572119903| lt 119896120572

119903

(119905) in (26) because there is no effective mechanismto guarantee it Motivated by [49] the command filter isemployed to constrain the magnitude of the virtual yaw rateby the time-varying constraint 119896120572

119903

(119905) The command filter isshown in Figure 3

Here the nominal virtual yaw rate 120572119903119888 is filtered toprovide the magnitude and bandwidth limited virtual yawrate 120572119903 and its derivative 119903 In addition an auxiliary systemwill be designed to compensate for the estimation errorΔ120572119903 =120572119903 minus 120572119903119888Step 1 (kinematics controller) Choose the following candi-date Lyapunov function 1198812

1198812 = 121205952119890 (28)

Taking the time derivative of (28) we obtain

2 = 120595119890 (119903119890 cos120601 + Δ120572119903 cos120601 + 120572119903119888 cos120601 minus 119889) (29)

To avoid the complex derivative computations of120595119889 includedin the nominal virtual control law 120572119903119888 a second-ordertracking-differentiator [50 51] will be used for the estimationof 119889 which is described as

V1 = V2

V2 = minus1198772 [1198861 (V1 minus 120595119889) + 1198862 V2119877 ] (30)

where 119877 1198861 and 1198862 are positive design constants 120595119889 isthe input signal of (30) V1 and V2 denote the respectiveestimations of 120595119889 and 119889 Let V2 = 119889119888 and define theestimation error as 120576 = 119889119888 minus 119889 (31)

According to Theorem 2 presented in [50] we havelim119877997888rarr+infin120576 = 0 which means that 120576 can be sufficiently smallby choosing a large 119877 Hence it can be concluded that thereexists arbitrarily small positive constant 120576119872 such that |120576| le 120576119872

As such the nominal virtual control law 120572119903119888 for (29) isgiven by

120572119903119888 = minus1198963120595119890 + 119889119888 minus 120599119890120594 + 1198881120577119903 minus (11988821 2) 120595119890cos120601 (32)


where 1198963 gt 12 and 1198881 gt 0 denote the design constants 120601 =plusmn1205872 120577119903 is the state of the auxiliary system and compensatesfor the constraint effect Δ120572119903 The auxiliary system is designedas

120577119903 = minus1198964120577119903 minus 1198911120577119903 + 1205741Δ120572119903 cos120601 10038161003816100381610038161205771199031003816100381610038161003816 gt 12058810 10038161003816100381610038161205771199031003816100381610038161003816 le 1205881 (33)

where 1198964 gt 1 and 1205741 gt 0 1205881 is a small positive design constantthat satisfies the performance requirement 1198911 is given by

1198911 = 1003816100381610038161003816120595119890Δ120572119903 cos1206011003816100381610038161003816 + (12) 12057421 (Δ120572119903 cos120601)2100381610038161003816100381612057711990310038161003816100381610038162 (34)

Remark 11 Note that in (33) 120577119903 will exponentially convergeonto the small set |120577119903| le 1205881 if Δ120572119903 = 0 If the state 120577119903 in (33)satisfies |120577119903| gt 1205881 then saturation effects occur again and theinitial value of the auxiliary system should be properly resetto ensure that 120577119903 can respond to the case Δ120572119903 = 0

Consider the candidate Lyapunov function

1198813 = 1198812 + 121205772119903 (35)

Invoking (32) and (33) the time derivative of 1198813 is given by

3 = 120595119890 ((119903119890 + Δ120572119903) cos120601 minus 1198963120595119890 + (119889119888 minus 119889) minus 120599119890120594+ 1198881120577119903 minus 119888212 120595119890) + 120577119903 (minus1198964120577119903 minus 1198911120577119903 + 1205741Δ120572119903 cos120601)= 120595119890119903119890 cos120601 + 120595119890Δ120572119903 cos120601 + 120595119890120576 minus 120599119890120595119890120594 + 1198881120595119890120577119903minus 11989631205952119890 minus 11989641205772119903 minus 1003816100381610038161003816120595119890Δ120572119903 cos1206011003816100381610038161003816 minus 1212057421 (Δ120572119903 cos120601)2minus 119888212 1205952119890 + 1205741120577119903Δ120572119903 cos120601

(36)

Note that in (36)

120595119890120576 le 121205952119890 + 1212057621198721198881120595119890120577119903 le 12119888211205952119890 + 121205772119903

1205741120577119903Δ120572119903 cos120601 le 1212057421 (Δ120572119903 cos120601)2 + 121205772119903 (37)

Then we have

3 le minus(1198963 minus 12)1205952119890 minus (1198964 minus 1) 1205772119903 minus 120599119890120595119890120594 + 120595119890119903119890 cos120601+ 121205762119872 le minus12058921198813 minus 120599119890120595119890120594 + 120595119890119903119890 cos120601 + 121205762119872

(38)

where 1205892 = min1198963 minus 12 1198964 minus 1Step 2 (dynamics controller) Consider the following time-varying symmetric BLF

1198814 = 12 log 1198962119903119890

(119905)1198962119903119890

(119905) minus 1199032119890 (39)

where 119896119903119890

(119905) R+ 997888rarr R+ Recalling Remark 10 the con-strained yaw rate controller will be proposed to guarantee theconstraint |119903119890| lt 119896119903

119890

(119905) Define the new variable as 120585119903 = 119903119890119896119903119890

(39) can be rewritten as

1198814 = 12 log 11 minus 1205852119903 (40)

It can be seen that 1198814 is positive definite and continuouslydifferentiable in the set |120585119903| lt 1 The time derivative of 1198814 iscalculated as

4 = 120599119903( 119903119890 minus 119903119890 119903119890119896119903119890

) (41)

with 120599119903 = 120585119903119896119903119890

(1 minus 1205852119903 )Substituting the third equation of (7) into (41) we have

4 = 120599119903(119891119903 + 120599T11990311205931199031 + 120599T11990321205931199032 + 120591119903 + 119889119903 minus 119903 minus 119903119890 119903119890119896119903119890

) (42)

The yaw rate control law for (42) becomes

120591119903 = minus119891119903 minus 120599T11990311205931199031 minus 120599T11990321205931199032 + 119903 minus 1198965119903119890 minus 1198965119903119890minus (1198962119903

119890

minus 1199032119890 ) 120595119890 cos120601 minus 119863119903 tanh(120599119903120590119903)(43)

where 1198965 = radic(119903119890

119896119903119890

)2 + 120603 with 120603 gt 0 a small constant1198965 gt 0 and 120590119903 gt 0 are design constants 119903 is directly obtainedfrom the output of the command filter 120599T1199031 120599T1199032 and 119863119903 areadaptive estimations of 120599T1199031 120599T1199032 and119863119903 respectively

The adaptation laws for 1205991199031 1205991199032 and 119863119903 are designed asfollows 1205991199031 = 1205991199031205931199031 minus 119896120599112059911990311205991199032 = 1205991199031205931199032 minus 11989612059921205991199032119863119903 = 120599119903 tanh(120599119903120590119903) minus 1198966119863119903

(44)

where 1198961205991 1198961205992 and 1198966 are positive design constantsNext we construct the candidate Lyapunov function as

follows

1198815 = 1198814 + 1212059911987911990311205991199031 + 1212059911987911990321205991199032 + 121198632119903 (45)

Taking the time derivative of (45) and using (42) and (44) weobtain

5 = 4 minus 1205991198791199031 1205991199031 minus 1205991198791199032 1205991199032 minus 119863119903 119863119903le minus1198965 12058521199031 minus 1205852119903 minus (1198965 + 119903

119890119896119903119890

)119903119890120599119903 + 1198961205991120599T11990311205991199031+ 1198961205992120599T11990321205991199032 + 120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) + 1198966119863119903119863119903minus 120595119890119903119890 cos120601

(46)


Note that

1198965 + 119903119890119896119903119890

ge 10038161003816100381610038161003816119896510038161003816100381610038161003816 minus 1003816100381610038161003816100381610038161003816100381610038161003816 1199031198901198961199031198901003816100381610038161003816100381610038161003816100381610038161003816 gt 0 (47)

120599T11990311205991199031 le 1003817100381710038171003817120599119903110038171003817100381710038172 minus 1003817100381710038171003817100381712059911990311003817100381710038171003817100381722 120599T11990321205991199032 le 1003817100381710038171003817120599119903210038171003817100381710038172 minus 1003817100381710038171003817100381712059911990321003817100381710038171003817100381722 119863119903119863119903 le 1198632119903 minus 11986321199032

(48)

and

120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) le 02785120590119903119863119903 (49)

Substituting the inequalities (47)-(49) and (2) into (46) in theset |120585119903| lt 1 we can obtain that

5 le minus211989651198814 minus (1198965 + 119903119890119896119903119890

) 12058521199031 minus 1205852119903 minus 11989612059912 100381710038171003817100381710038171205991199031100381710038171003817100381710038172minus 11989612059922 100381710038171003817100381710038171205991199032100381710038171003817100381710038172 minus 11989662 1198632119903 + 120583119903 minus 120595119890119903119890 cos120601

le minus211989651198814 minus 11989612059912 100381710038171003817100381710038171205991199031100381710038171003817100381710038172 minus 11989612059922 100381710038171003817100381710038171205991199032100381710038171003817100381710038172 minus 11989662 1198632119903 + 120583119903minus 120595119890119903119890 cos120601 le minus12058931198815 + 120583119903 minus 120595119890119903119890 cos120601(50)

where 1205893 = min21198965 1198961205991 1198961205992 1198966 with 120583119903 a positive constantdefined as

120583119903 = 11989612059912 1003817100381710038171003817120599119903110038171003817100381710038172 + 11989612059922 1003817100381710038171003817120599119903210038171003817100381710038172 + 11989662 1198632119903 + 02785120590119903119863119903 (51)

33 Velocity Tracking Control In this subsection thedesigned control force 120591119906 should enforce the followingconstraint on the velocity lim119905997888rarrinfin119906(119905) 997888rarr 119906119889(119905) Inaddition the constraint |119906119890| lt 119896119906

119890

should also be strictlyguaranteed Using the first equation in (7) the time derivativeof 119906119890 is calculated as119890 = V119903 + 119891119906 + 120599T11990611205931199061 + 120599T11990621205931199062 + 120591119906 + 119889119906 minus 119889 (52)Choosing the barrier Lyapunov function

1198816 = 12 log 11989621199061198901198962119906

119890

minus 1199062119890 (53)

and differentiating (53) yield6 = 120599119906119890 (54)

where 120599119906 = 119906119890(1198962119906119890

minus 1199062119890)We design the surge velocity tracking controller for 120591119906 as

follows

120591119906 = minusV119903 minus 120599T11990611205931199061 minus 120599T11990621205931199062 minus 119863119906 tanh(120599119906120590119906) minus 119891119906minus 1198967119906119890 + 119889 minus 120599119890 (1198962119906119890

minus 1199062119890) sin (120595 minus 120574119901) (55)

where 1198967 gt 0 and 120590119906 gt 0 are design constants

The adaptation laws for 1205991199061 1205991199062 and119863119906 are given as1205991199061 = 1205991199061205931199061 minus 119896120599312059911990611205991199062 = 1205991199061205931199062 minus 11989612059941205991199062119863119906 = 120599119906 tanh(120599119906120590119906) minus 1198968119863119906(56)

where 1198961205993 1198961205994 and 1198968 are positive design constantsConstruct the Lyapunov function 1198817 as

1198817 = 1198816 + 12120599T11990611205991199061 + 12120599T11990621205991199062 + 121198632119906 (57)

Taking the time derivative of (57) and invoking (54)-(56) weobtain

7 = 6 minus 120599T1199061 1205991199061 minus 120599T1199062 1205991199062 minus 119863119906 119863119906le minus1198967 11990621198901198962119906

119890

minus 1199062119890 + 1198961205993120599T11990611205991199061 + 1198961205994120599T11990621205991199062 + 1198968119863119906119863119906+ 120599119906 (119889119906 minus 119863119906 tanh(120599119906120590119906)) minus 120599119890119906119890 sin (120595 minus 120574119901)

(58)

Note also that 120599T11990611205991199061 le (12059911990612 minus 12059911990612)2 12059911987911990621205991199062 le(12059911990622 minus 12059911990622)2 and119863119906119863119906 le (1198632119906 minus 1198632119906)2We then have

7 le minus211989671198816 minus 11989612059932 100381710038171003817100381710038171205991199061100381710038171003817100381710038172 minus 11989612059942 100381710038171003817100381710038171205991199062100381710038171003817100381710038172 minus 11989682 1198632119906 + 120583119906minus 120599119890119906119890 sin (120595 minus 120574119901)le minus12058941198817 + 120583119906 minus 120599119890119906119890 sin (120595 minus 120574119901)

(59)

where 1205894 = min21198967 1198961205993 1198961205994 1198968 and120583119906 is a positive constantdefined as

120583119906 = 11989612059932 1003817100381710038171003817120599119906110038171003817100381710038172 + 11989612059942 1003817100381710038171003817120599119906210038171003817100381710038172 + 11989682 1198632119906 + 02578120590119906119863119906 (60)

Remark 12 In (55) the desired surge speed 119906119889(119905) is requiredto be continuously differentiable In previous works [10 1145] the desired surge speed 119906119889 was chosen as a constant forall 119905 gt 0 However in practice the desired surge speed of ahovercraft is often directly switched to another value whichwill lead to the nonexistence of 119889(119905) Thus to ensure that thedesired surge speed is differentiable when 119906119889 changes from1199061198891 to 1199061198892 between any time interval 119905119886 and 119905119887 we give a speedsmoother as follows119906119889 = 1199061198891 + ℎ (119905 119905119886 119905119887 120574119905) (1199061198892 minus 1199061198891) (61)

The scalar function ℎ(119905 119905119886 119905119887 120574119905) is defined as

ℎ (119905 119905119886 119905119887 120574119905) = 119891 (120591)119891 (120591) + 120574119905119891 (1 minus 120591) (62)

where 120591 = (119905minus119905119886)(119905119887minus120574119905)119891(120591) = 0 if 120591 le 0 and119891(120591) = 119890minus1120591if 120591 gt 0 with 119905119886 lt 119905119887 and 120574119905 a positive constant The speedsmoother is shown in Figure 4


d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu

Figure 4 The desired surge speed and its derivative

34 Stability Analysis

Theorem 13 Consider the parameterized hovercraft models(7) and (10) in the presence of parametric uncertainties andexternal disturbances and assume that Assumptions 5ndash8 aresatisfied If the update law for the path variable 120596 is chosen as(22) the desired heading angle of the hovercraft is calculatedby (23) the auxiliary system is designed as (33) the adaptationlaws for unknown parameters and external disturbances aregiven by (44) and (56) and the controllers are obtained from(43) and (55) then the following properties hold

(i) All the error signals in the closed-loop system areuniformly ultimately bounded (UUB)

(ii) The static constraints on position and surge velocitytracking errors and the time-varying constraints on the yaw rateare never violated ie |120585119904| lt 119896119904 |120585119890| lt 119896119890 |119906119890| lt 119896119906

119890

and|119903(119905)| lt 119896119903(119905) hold for all 119905 gt 0Proof (i) Construct the following candidate Lyapunov func-tion 119881 = 1198811 + 1198813 + 1198815 + 1198817 (63)

By virtue of (24) (38) (50) and (59) the time derivative of119881is given by

le minus12058911198811 minus 12058921198813 minus 12058931198815 minus 12058941198817 + 120599119890120595119890120594 + 120583119903+ 120599119890119906119890 sin (120595 minus 120574119901) minus 120599119890120595119890120594 + 120595119890119903119890 cos120601minus 120595119890119903119890 cos120601 minus 120599119890119906119890 sin (120595 minus 120574119901) + 121205762119872 + 120583119906

le minus12058911198811 minus 12058921198813 minus 12058931198815 minus 12058941198817 + 120583119906 + 120583119903 + 121205762119872le minus120589119881119881 + 120583119881

(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872

If the initial conditions satisfy Assumption 6 then theconstructed Lyapunov function 119881 satisfies the inequality

119881 (119905) le Γ (119905) = (119881 (0) minus 120583119881120589119881 ) exp (minus120589119881119905) + 120583119881120589119881 (65)

From (65) we know that lim119905997888rarrinfin119881(119905) le lim119905997888rarrinfinΓ(119905) =120583119881120589119881 Therefore all the error signals in the closed-loopsystem are uniformly ultimately bounded Noting the expres-sions of 120583119881 and 120589119881 the ultimate bounds can be madearbitrarily small by adjusting the design parameters

(ii) forall119905 gt 0 the BLF (14) (40) and (53) satisfy thefollowing inequalities

12 log 11989621198941198962119894 minus 1205852119894 le 119881 (119905) le Γ (119905)12 log 1198962119906

1198901198962119906119890

minus 1199062119890 le 119881 (119905) le Γ (119905)12 log 11 minus 1205852119903 le 119881 (119905) le Γ (119905)(66)

where 119894 = 119904 119890Then from (66) we obtain10038161003816100381610038161205851199041003816100381610038161003816 le 119896119904radic1 minus exp (minus2Γ (119905)) lt 11989611990410038161003816100381610038161205851198901003816100381610038161003816 le 119896119890radic1 minus exp (minus2Γ (119905)) lt 11989611989010038161003816100381610038161199061198901003816100381610038161003816 le 119896119906119890radic1 minus exp (minus2Γ (119905)) lt 119896119906

11989010038161003816100381610038161205851199031003816100381610038161003816 le radic1 minus exp (minus2Γ (119905)) lt 1(67)

In (67) from |120585119903| lt 1 forall119905 gt 0 we know that |119903119890(119905)| lt119896119903119890

(119905) forall119905 gt 0 Recalling Remark 10 we conclude that|119903(119905)| lt 119896119903(119905) Therefore all the prescribed constraints ofthe hovercraft system can be guaranteed for forall119905 gt 0 Thiscompletes the proof

4 Simulation

In this section numerical simulations will be implementedto demonstrate the effectiveness and safety of the proposedcontrol scheme for hovercraft navigation and to compareour scheme with the results presented in [11 45] A generalcurvilinear path simulation scenario is considered In simu-lations the uncertainty parameters in (7) including 119878119909119886 119878119910119886119878119911119886 119878ℎ and 119862

lowast have 20 maximum relative uncertaintiesaccording to their nominal values ie their real values arechosen according to

119878lowast = 119878lowastnominal + 120582119878lowastnominal119862lowast = 119862

lowastnominal + 120582119862lowastnominal (68)

where 120582 is randomly selected from [minus02 02] The nominalparameters of the hovercraft are shown in Appendix A


Table 1 Nominal values of the hovercraft in the simulation

Parameter Nominal Value SI-Unit Parameter Nominal Value SI-Unit1198621199091198860 -0345 - 119862119910ℎ0 -000011 -119862120575119903119909119886 -000516 1rad 119862120601119910ℎ -00012 1rad119862120573119909119886 -02504 1rad 119862120573119910ℎ 004582 1rad119862 120573119909119886 -00001 srad 1198621198981199111198860 -0000356 -119862119909ℎ0 -01049 - 119862120575119903119898119911119886 -0001534 1rad119862120601119909ℎ 000014 1rad 119862120573119898119911119886 -005412 1rad119862120573119909ℎ 0191 1rad 119862 120573

119898119911119886 -000021 srad1198621199101198860 -0001 - 119862120601119898119911ℎ 0 1rad119862120575119903119910119886 0002614 1rad 119862120573119898119911ℎ 0 1rad119862120573119910119886 01645 1rad 119898 40000 kg119862 120573119910119886 000042 srad 119868119909 247810 kgm2119878119909119886 4527 m2 119868119911 1746347 kgm2119878119910119886 93 m2 119878119911119886 268 m2119878ℎ 21 m2

4

5

u [ms]

r [de

gs]

Yaw rate constraints

8 10

3

212 14

45

Figure 5 The yaw rate constraints on the hovercraft

Table 1 For numerical simulations the time-varyingbounded disturbances in (7) are generated by using thefirst-order Gauss-Markov process [52]119889119906 + 1198861119889119906 = 1205961119889V + 1198862119889V = 1205962119889119903 + 1198863119889119903 = 1205963119889119901 + 1198864119889119901 = 1205964

(69)

where 120596119894 (119894 = 1 2 3 4) are zero-mean Gaussian white noiseprocesses and 119886119894 ge 0 (119894 = 1 2 3 4) are constants

To demonstrate that the control objective can be achievedby the proposed controller we first set the constant con-straints on position and surge velocity to be 119896119904 = 30m 119896119890 =30m and 119896119906

119890

= 1ms The time-varying yaw rate constraintin our simulations is shown in Figure 5 which is similar to

the yaw rate constraint in [33] We can see that the yaw rateconstraint of the hovercraft decreases with increasing surgespeed To adapt to practical applications it is determined thatthe yaw rate should be further restricted to guarantee thenavigation safety of a hovercraft with a higher speedThe yawrate constraint 119896119903(119905) can be written as 119896119903(119905) = minus025119906(119905) + 7with 119906(119905) ge 8ms The maximum yaw rate tracking error islimited by |119903119890| le 1 degs Thus the time-varying magnitudeof the limiter in Figure 3 is set as 119896120572

119903

(119905) = minus025119906(119905) + 6with 119906(119905) ge 8ms and the yaw rate tracking error constraintsshould satisfy the conditions 119896119903

119890

(119905) le 119896119903(119905)minus119896120572119903

(119905) = 1 degsThe control parameters are listed in Table 2

In addition to our proposed controller we simulate twodifferent controllers for comparison which are assigned asfollows

Case 1 This comparative controller is derived from [11]by using traditional adaptive LOS guidance The yaw rateconstraint is not considered

Case 2 This comparative controller is designed by using thestrategy in [45] which can only guarantee the position con-straints However the yaw rate constraint is not considered

Case 3 This controller in this case is our proposed controlscheme

The desired path for a curvilinear path following simu-lation is given as 119883119901 = [120596 2000 sin(120596550)]T m The initialpositions of the hovercraft are chosen as (119909(0) 119910(0) 120595(0)) =(29m 29m 0 rad) the initial velocities are (119906(0) V(0)) =(75ms 0ms) and 119903(0) = 00087 rads The initial statesof the hovercraft satisfy Assumption 6 and according to (61)the desired surge speed 119906119889(119905) varies from 8 ms to 10 msbetween 119905119886 = 406 s and 119905119887 = 410 s For comparison purposesthe desired surge speed in Cases 1 and 2 is directly switchedfrom 8 ms to 10 ms at 119905119904 = 408 s To analyse the control per-formances under these three controllers simulation resultsare given as follows

The path following results in Figure 6 demonstrate thatthe proposed controller has improved performance in terms


Table 2 Control parameters in hovercraft simulations

Parameter Value Parameter Value1198961 2 120590119906 011198962 80 120590119903 011198963 1 1198961205991 51198881 01 1198961205992 31198964 15 1198961205993 151198965 001 1198961205994 41198965 3 1198967 11198966 20 1198968 02

desired pathproposed method

Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]

Figure 6 Curvilinear path following

of the tracking accuracy and convergence rate Figure 7 showsthat the path following error constraints cannot be violatedunder the proposed controller Neither the along-track errornor the cross-track error exceeds the position constraints119896119904 and 119896119890 However for the Case 1 method the positiontracking errors transgress the predefined constraints Theposition constraint problem is also handled by the Case 2method in which the path following errors are also regulatedwithin the prescribed bounds Figure 8 shows the surge speedtracking results under the different control schemes Wecan see that the surge speed smoothly and continuouslytracks the desired surge speed 119906119889 under the proposed speedcontroller However the surge speed under Cases 1 and 2 isdiscontinuous and has a large overshoot

The yaw rate tracking results under the three controllersare presented in Figure 9 When 119905 lt 119905119904 = 408 s the desiredsurge speed is 119906119889 = 8ms thus the yaw rate constraint iscalculated as 119896119903 = 5 degs and the virtual yaw rate constraintis 119896120572

119903

= 4 degs After 119905119904 119906119889 increases to 10 ms thereforethe yaw rate constraint is calculated as 119896119903 = 45 degs and119896120572119903

= 35 degs The yaw rate tracking error constraint is setas 119896119903

119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]

proposed methodCase 1 Case 2

0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]

Constraint is violated


Figure 7 Position errors of curvilinear path following


Case 1 Case 2 proposed method

406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]

Figure 8 The surge speed of the curvilinear path following

minus5

0

5

r [de

gs]

r under proposed methodr under proposed method

minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]

r under Case 2r under Case 2


100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

Figure 9 The yaw rate of the curvilinear path following

In the first subfigure of Figure 9 we observe that themagnitude of the virtual yaw rate 120572119903(119905) is limited by thedesigned command filter The yaw rate of the hovercraft has

0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]

Case 1Case 2proposed method

0 10 20 30minus1

0

1

Constraint isviolated

Figure 10 The yaw rate tracking errors of the curvilinear pathfollowing

smaller tracking errors in both transient and steady-stateperformances as expected However in the second and thirdfigures of Figure 10 it can be observed that the yaw ratetracking errors have large transient and steady-state errorThe yaw rate tracking errors under the three controllers areillustrated in Figure 10 The dotted line denotes the yaw rateconstraint 119896119903

119890

(119905) It is shown that the yaw rate constraints areguaranteed under the proposed controller which indicatesthat the yaw rate of the hovercraft can track the desired yawrate with the prescribed precision Therefore the yaw rateof the hovercraft will never violate the corresponding safetyboundary In contrast the yaw rate tracking errors underthe methods of Cases 1 and 2 both violate the prescribedconstraints because the yaw rate tracking error constraintcannot be guaranteed by using the conventional quadraticLyapunov function in theory Due to the page limitation inFigure 11 the estimate values of the uncertainty parametersin control laws (43) and (55) are shown in the form 1205851 =119898120599T11990621205931199062 1205852 = 119868119911120599T11990321205931199032 1205853 = 119898120599T11990611205931199061 and 1205854 = 119868119911120599T11990311205931199031Figure 12 shows the estimate values 119863119906 and 119863119903 The controlinputs 120591119906 and 120591119903 are shown in Figure 13 we see that thecontrol signals under our proposed method perform betterthan another two controllers in terms of oscillations

5 Conclusion

A constrained adaptive path following control approach hasbeen proposed for a hovercraft under parametric uncertain-ties and external disturbances It is shown that the proposedcontroller can strictly ensure the satisfaction of the prescribedposition and velocity constraints by resorting to the barrierLyapunov function design method The command filter witha time-varying magnitude limit is employed to generate the


x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

Figure 11 Estimated values of the uncertain parameters

＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ

Figure 12 Estimated values of the external disturbances

desired virtual yaw rate and to reduce the computationalcomplexity while an auxiliary system is designed to com-pensate for the estimation error The uncertain parameters

0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2

ＯFigure 13 Control inputs of the curvilinear path following

and the upper bounds of the external disturbances arewell estimated by designing parameter adaptation lawsSimulations have indicated the superiority of the proposedcontrol scheme Because the surge speed of the hovercraftis usually high the time-varying delay of the measurementsmay happen In our future work we will take into account thetime delays in the feedback loop

Appendix

A Nominal Values of the Hovercraft Model

The nominal values of the hovercraft are given in Table 1

B Expressions of the Hovercraft Model

The expressions of parameters and functions in (7) are givenas follows

119891119906 = minus120588119886119881119886119876 cos120573119898 119891V = (minus120588119886119881119886119876 sin120573 minus 120588119886119876119888 (2119901119888120588119886)12)119898 119891119903 = 1119868119911 (minus120588119886119881119886119876 sin120573119897119866

119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )

12 119911119892minus 120588119886119881119886119876 sin120573 (119911119898 minus 119911119892) minus119882ℎ tan120601)


1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T

1205931199061 = 119902119886119898 [1 120575119903 120573 120573]T where 119902119886 = 121205881198861198812119886 1205991199062 = 119878ℎ [1198620119909ℎ 119862120601119909ℎ 119862120573119909ℎ]T 1205931199062 = 119902ℎ119898 [1 120601 120573]T where 119902ℎ = 12120588ℎ1198802120599V1 = 119878119910119886 [1198621199101198860 119862120575119903119910119886 119862120573119910119886 119862 120573

119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability

The data in this paper is derived from the Internationalcooperation projects Hence the date is private

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under grant 51309062

References

[1] L Yun and A Bliault Theory amp Design of Air Cushion CraftButterworth-Heinemann Oxford UK 2000

[2] J A Fein A H Magnuson and D D Moran ldquoDynamicperformance characteristics of an air cushion vehiclerdquo Journalof Hydronautics vol 9 no 1 pp 13ndash24 1975

[3] R Hayashi K Osuka and T Ono ldquoTrajectory control of an aircushion vehiclerdquo in Proceedings of the IEEERSJ InternationalConference on Intelligent Robots and Systems rsquo94 (IROS rsquo94)rsquoAdvanced Robotic Systems and the Real World (IROSrsquo94) pp1906ndash1913 Munich Germany 1994

[4] M Breivik and T I Fossen ldquoPath following for marine surfacevesselsrdquo in Proceedings of the Oceanrsquo04 - MTSIEEE Techno-Oceanrsquo04 Bridges across the Oceans - Conference Proceedingsvol 4 pp 2282ndash2289 Japan November 2004

[5] Z Li J Sun and S Oh ldquoDesign analysis and experimentalvalidation of a robust nonlinear path following controller formarine surface vesselsrdquo Automatica vol 45 no 7 pp 1649ndash1658 2009

[6] S Oh and J Sun ldquoPath following of underactuated marine sur-face vessels using line-of-sight based model predictive controlrdquoOcean Engineering vol 37 no 2-3 pp 289ndash295 2010

[7] C Liu R RNegenborn X Chu andH Zheng ldquoPredictive pathfollowing based on adaptive line-of-sight for underactuatedautonomous surface vesselsrdquo Journal of Marine Science andTechnology (Japan) vol 23 no 3 pp 483ndash494 2018

[8] T I Fossen and A M Lekkas ldquoDirect and indirect adaptiveintegral line-of-sight path-following controllers formarine craftexposed to ocean currentsrdquo International Journal of AdaptiveControl and Signal Processing vol 31 no 4 pp 445ndash463 2017

[9] Y-X Li andG-H Yang ldquoModel-based adaptive event-triggeredcontrol of strict-feedback nonlinear systemsrdquo IEEETransactionson Neural Networks and Learning Systems vol 29 no 4 pp1033ndash1045 2018

[10] Z Zheng and L Sun ldquoPath following control for marine surfacevessel with uncertainties and input saturationrdquo Neurocomput-ing vol 177 pp 158ndash167 2016

[11] L Liu DWang and Z Peng ldquoPath following of marine surfacevehicles with dynamical uncertainty and time-varying oceandisturbancesrdquo Neurocomputing vol 173 pp 799ndash808 2016

[12] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015

[13] H Sira-Ramırez and C A Ibanez ldquoOn the control of thehovercraft systemrdquo Dynamics and Control An InternationalJournal vol 10 no 2 pp 151ndash163 2000

[14] A P Aguiar L Cremean and J P Hespanha ldquoPosition trackingfor a nonlinear underactuated hovercraft Controller designand experimental resultsrdquo in Proceedings of the 42nd IEEEConference on Decision and Control pp 3858ndash3863 December2003

[15] I Fantoni R Lozano F Mazenc and K Y Pettersen ldquoStabiliza-tion of a nonlinear underactuated hovercraftrdquo in Proceedings ofthe 38th IEEE Conference on Decision and Control (CDC) vol3 pp 2533ndash2538 December 1999

[16] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002

[17] Y Han and X Liu ldquoHigher-order sliding mode control fortrajectory tracking of air cushion vehiclerdquo Optik - InternationalJournal for Light and Electron Optics vol 127 no 5 pp 2878ndash2886 2016

[18] J Tsinias and I Karafyllis ldquoISS property for time-varyingsystems and application to partial-static feedback stabilization


and asymptotic trackingrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 44 no 11 pp2179ndash2184 1999

[19] S S Pavlichkov S N Dashkovskiy and C K Pang ldquoUniformstabilization of nonlinear systems with arbitrary switchings anddynamic uncertaintiesrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 62 no 5 pp2207ndash2222 2017

[20] H Wu ldquoSimple adaptive robust output tracking controlschemes of uncertain parametric strict-feedback non-linearsystems with unknown input saturationsrdquo IET Control Theoryamp Applications vol 12 no 12 pp 1694ndash1703 2018

[21] A Astolfi D Karagiannis and R Ortega Nonlinear and Adap-tive Control with Applications Springer London UK 2008

[22] M Krstic I Kanellakopoulos and P Kokotovic ldquoAdaptivenonlinear control without overparametrizationrdquo Systems ampControl Letters vol 19 no 3 pp 177ndash185 1992

[23] G Zhang X Zhang and Y Zheng ldquoAdaptive neural path-following control for underactuated ships in fields of marinepracticerdquo Ocean Engineering vol 104 pp 558ndash567 2015

[24] G Zhang and X Zhang ldquoA novel DVS guidance principle androbust adaptive path-following control for underactuated shipsusing low frequency gain-learningrdquo ISA Transactions vol 56pp 75ndash85 2015

[25] J Shin D J Kwak and Y-I Lee ldquoAdaptive path-followingcontrol for an unmanned surface vessel using an identifieddynamic modelrdquo IEEEASME Transactions on Mechatronicsvol 22 no 3 pp 1143ndash1153 2017

[26] T I Fossen and K Y Pettersen ldquoOn uniform semiglobalexponential stability (USGES) of proportional line-of-sightguidance lawsrdquo Automatica vol 50 no 11 pp 2912ndash2917 2014

[27] K Y Pettersen ldquoLyapunov sufficient conditions for uniformsemiglobal exponential stabilityrdquo Automatica vol 78 pp 97ndash102 2017

[28] A M Lekkas and T I Fossen ldquoIntegral LOS path followingfor curved paths based on a monotone cubic hermite splineparametrizationrdquo IEEE Transactions on Control Systems Tech-nology vol 22 no 6 pp 2287ndash2301 2014

[29] T I Fossen K Y Pettersen and R Galeazzi ldquoLine-of-sight pathfollowing for dubins paths with adaptive sideslip compensationof drift forcesrdquo IEEE Transactions on Control Systems Technol-ogy vol 23 no 2 pp 820ndash827 2015

[30] L Liu D Wang Z Peng and T Li ldquoModular adaptive con-trol for LOS-based cooperative path maneuvering of multipleunderactuated autonomous surface vehiclesrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 47 no 7 pp1613ndash1624 2017

[31] L Liu D Wang and Z Peng ldquoESO-based line-of-sight guid-ance law for path following of underactuated marine surfacevehicles with exact sideslip compensationrdquo IEEE Journal ofOceanic Engineering vol 42 no 2 pp 477ndash487 2017

[32] J Miao S Wang M M Tomovic and Z Zhao ldquoCompoundline-of-sight nonlinear path following control of underactuatedmarine vehicles exposed to wind waves and ocean currentsrdquoNonlinear Dynamics vol 89 no 4 pp 2441ndash2459 2017

[33] RGranG Carpenter andRWKlein ldquoComputer aided designof a control system for a hovercraftrdquo in Proceedings of the 21stIEEE Conference on Decision and Control vol 21 pp 1347ndash13531982

[34] D Q Mayne J B Rawlings C V Rao and P O M ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[35] M Krstic and M Bement ldquoNonovershooting control of strict-feedback nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 51 no 12 pp 1938ndash1943 2006

[36] Z Zheng and M Feroskhan ldquoPath following of a surfacevessel with prescribed performance in the presence of inputsaturation and external disturbancesrdquo IEEEASMETransactionson Mechatronics vol 22 no 6 pp 2564ndash2575 2017

[37] K P Tee and S S Ge ldquoControl of nonlinear systems withfull state constraint using a barrier lyapunov functionrdquo inProceedings of the 48th IEEEConference onDecision and ControlHeld Jointly with 28th Chinese Control Conference (CDCCCCrsquo09) pp 8618ndash8623 Shanghai China December 2009

[38] B Niu D Wang H Li X Xie N D Alotaibi and FE Alsaadi ldquoA novel neural-network-based adaptive controlscheme for output-constrained stochastic switched nonlinearsystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 49 no 2 pp 418ndash432 2019

[39] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[40] C Wang Y Wu and J Yu ldquoBarrier Lyapunov functions-baseddynamic surface control for pure-feedback systems with fullstate constraintsrdquo IET ControlTheory amp Applications vol 11 no4 pp 524ndash530 2017

[41] B Niu and J Zhao ldquoBarrier Lyapunov functions for the outputtracking control of constrained nonlinear switched systemsrdquoSystems amp Control Letters vol 62 no 10 pp 963ndash971 2013

[42] H An H Xia and C Wang ldquoBarrier Lyapunov function-based adaptive control for hypersonic flight vehiclesrdquoNonlinearDynamics vol 88 no 3 pp 1833ndash1853 2017

[43] X Wang Y Zhang and H Wu ldquoSliding mode control basedimpact angle control guidance considering the seekers field-of-view constraintrdquo ISA Transactions vol 61 pp 49ndash59 2015

[44] X Jin ldquoAdaptive fault tolerant control for a class of inputand state constrained MIMO nonlinear systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 26 no 2 pp 286ndash302 2016

[45] Z Zheng L Sun and L Xie ldquoError-constrained los pathfollowing of a surface vessel with actuator saturation and faultsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 48 no 10 pp 1794ndash1805 2018

[46] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquo Automatica vol 47 no11 pp 2511ndash2516 2011

[47] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011

[48] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control John Wiley amp Sons New York NY USA 2011

[49] J A Farrell M Polycarpou M Sharma and W Dong ldquoCom-mand filtered backsteppingrdquo IEEE Transactions on AutomaticControl vol 54 no 6 pp 1391ndash1395 2009

[50] X Wang Z Chen and Z Yuan ldquoDesign and analysis fornew discrete tracking-differentiatorsrdquo Applied Mathematics-AJournal of Chinese Universities Series B vol 18 no 2 pp 214ndash222 2003


[51] X Bu X Wu M Tian J Huang R Zhang and Z Ma ldquoHigh-order tracking differentiator based adaptive neural control ofa flexible air-breathing hypersonic vehicle subject to actuatorsconstraintsrdquo ISA Transactions vol 58 pp 237ndash247 2015

[52] T I Fossen ldquoHow to incorporate wind waves and oceancurrents in themarine craft equations ofmotionrdquo inProceedingsof the 9th IFAC Conference on Manoeuvring and Control ofMarine Craft vol 45 no 27 pp 126ndash131 2012

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Submit your manuscripts atwwwhindawicom


structure and parametric uncertainties for control design Anexceedingly simplified hovercraft model was derived in [13ndash15] in which the hydrodynamic damping coefficients wereassumed to be zero The same hovercraft model was adoptedin [16 17] However because the hydrodynamic force andmoment are not included in this model the proposedmodel-based controllers might not achieve the desired controlobjective in practical applications Furthermore a curve-fitted hovercraft model was obtained in [1] by replacing thecomplex hydrodynamic and aerodynamic force and momentfunctionswith curve-fitted approximations by neglecting theheave and pitch motion a four-degree-of-freedom (DOF)control-oriented hovercraft model was obtained for controldesign However the effects of parametric uncertaintiesand external disturbances were not considered in [1] Theuncertainties and disturbances always exist in a practicalcontrol system and some works have investigated the feed-back control schemes for different form uncertain nonlinearsystems such as triangular form systems [18 19] and strict-feedback systems [20] In addition the adaptive controlmethod is regarded as a powerful method to cope withsuch system uncertainties [21] For example [22] presenteda backstepping-based adaptive control scheme to estimatethe unknown parameters and the tuning-function basedapproach was proposed to avoid the overparametrization inbackstepping design The adaptive backstepping techniquehas been widely applied in path following control of vessels[23ndash25] To improve the tracking performance of a hovercraftunder these system uncertainties the above 4-DOF hover-craft dynamics [1] are rewritten into a parameterized formin this paper and the adaptive control method is adopted toestimate the uncertain parameters and external disturbances

LOS guidance which has been widely explored forsurface vessels due to its simplicity and small computationalburden is an effective guidance algorithm for path followingcontrol In [26 27] it was shown that the equilibrium pointof the proportional LOS guidance law is uniformly andexponentially semi-globally stable To compensate for thevehiclersquos constant sideslip due to environmental disturbancesan integral LOS (ILOS) approach was proposed in [28 29]by adding an additive integral action into the conventionalLOS guidance In [10 30] an adaptive LOS (ALOS)-basedcontroller was developed with a parameter adaption law toeliminate the effects of system uncertainties In [31] theauthors discussed the drawbacks of the ILOS and ALOSA modified extended state observer- (ESO-) based LOSguidance approachwas proposed to identify the time-varyingsideslip angle Reference [32] presented a compound LOSguidance by combining the time delay control and ESOtechnique to estimate the time-varying ocean currents Notethat all the aforementioned LOS guidance algorithms onlyregulate the steady state of position tracking errors to bezero while the transient tracking performance cannot beguaranteed However it is important to ensure the prescribedtransient tracking performance of a hovercraft in practiceMoreover the yaw rate of a hovercraft should be limited tobelow the stability boundary for safe navigation [33] If themaximum yaw rate of a hovercraft exceeds the correspondingstability boundary for a period of time the hovercraft will

become unstable especially at high speeds which can evenlead to a capsizing hazard As shown in [33] the stabilityboundary of the yaw rate varies with the surge speed ofa hovercraft Thus the yaw rate constraint should also betime varying because the desired speed command for ahovercraft can change in practice To the authorrsquos knowledgethe problem of tracking error-constrained path followingcontrol for an uncertain hovercraft with a time-varying yawrate constraint has rarely been considered

To guarantee the state or output constraints of thesystem some control methods have been included in modelpredictive control [34] nonovershooting control [35] andprescribed performance control [36] However in thesemethods the state constraints remain difficult to guaranteein practical applications More recently a barrier Lyapunovfunction (BLF) has been proposed for nonlinear systems toensure the state and output constraints [37 38] which hasbeen applied to strict-feedback systems with output con-straints [39] pure-feedback systems [40] switched systems[41] and practical applications for hypersonic flight vehicles[42] or missile guidance [43] In addition to the conventionallogarithmic function form a modified tan-type BLF wasproposed in [44] which is a general method for systems withstate constraints because it works even if the state constraintsare removed In [45] the authors utilized the tan-type BLFto design an error-constrained LOS path following controllerfor a 3-DOF conventional surface vessel To break the staticconstraint limitations in the abovementioned works a time-varying output constraint was handled in [46] by using atime-varying BLF which allowed the output constraints tobe both time varying and asymmetric Moreover to facilitatethe backstepping design for the attitude subsystem of ahovercraft a command filter was introduced to avoid theanalytical computation of the virtual control laws [47]

Motivated by the above considerations a BLF-basedadaptive path following control scheme is developed for a 4-DOF underactuated hovercraft subject to parametric uncer-tainties and external disturbancesThemain contributions ofthis paper can be summarized as follows(1) A novel position-constrained LOS guidance algo-rithm is proposed by introducing BLF into the classical LOSguidance design procedure The proposed LOS guidance canguarantee the prescribed tracking performance in terms ofposition tracking errors(2) By virtue of the time-varying BLF an attitude con-troller is proposed to ensure the yaw rate time-varyingconstraints of a hovercraft The yaw rate of a hovercraftwill not exceed the stability boundary which has practicalsignificance for the safe navigation of a hovercraft(3) A command filter is integrated into the controldesign to generate the amplitude-constrained virtual controllaws which avoid the analytic computation of the virtualcontrol law derivative In addition a novel auxiliary systemis developed to compensate for the filtering error

The remainder of this paper is organized as follows Theproblem formulation is introduced in Section 2 Section 3 isdevoted to the LOS guidance altitude subsystem and velocitysubsystem control design Numerical simulation examplesare presented in Section 4 Section 5 concludes the paper


xE

OE yE

zE

(sway)

yb

u((surge)p(roll)

xbr(yaw)

zb

Ob



21 Preliminaries




0 le 10038161003816100381610038161205781003816100381610038161003816 minus 120578 tanh(120578120575) le 02785120575 (1)






= V119903 + (119865119909119886 + 119865119909ℎ + 119865119909119899 + 119865119909119879)119898V = minus119906119903 + (119865119910119886 + 119865119910ℎ + 119865119910119888 + 119865119910119899)119898119903 = (119872119911119886 +119872119911ℎ +119872119911119888 +119872119911119899 +119872119911120575)119868119911

= (119872119909119886 +119872119909ℎ +119872119909119888 +119872119909119899 +119872119909119866)119868119909

(3)


119865119909119886 = 051205881198861198812119886 119878119909119886 (1198621199091198860 + 119862120575119903119909119886120575119903 + 119862120573119909119886120573 + 119862 120573119909119886

120573)119865119910119886 = 051205881198861198812119886 119878119910119886 (1198621199101198860 + 119862120575119903119910119886120575119903 + 119862120573119910119886120573 + 119862 120573

119910119886120573)

119872119911119886 = 051205881198861198812119886 119878119911119886119897119886 (1198621198981199111198860 + 119862120575119903119898119911119886120575119903 + 119862120573119898119911119886120573 + 119862 120573119898119911119886

120573)119872119909119886 = 119865119910119886119911119886

(4)



(5)







(6)



(7)



120574119901 (120596) = atan2 (1199101015840119901 (120596) 1199091015840119901 (120596)) (8)


North

East Xp

pxp

s

yp

e

u

U

X




T [119909 minus 119909119901119910 minus 119910119901] (9)


[[[[[[

120601]]]]]]= [[[[[[


]]]]]][[[[[[

119906V119903119901]]]]]]

(10)





]]]]]+ [[




(12)






Suppose that 119896119904 119896119890 and 119896119906119890


119890


119890




119890









1 = 120599119904 120585119904 + 120599119890 120585119890 (15)




(16)




with



120595119890 = 120595 minus 120595119889 (20)



(21)











(24)

where


sin120595119890120595119890 (25)



119890997888rarr0(sin120595119890120595119890) = 1




where 119896120572119903


where 119896119903119890


Magnitude limiter

rckr

(t)

-kr(t)

+ +

- -

2n

2nn2nn

1

s

1

s

r







119903


119903



1198812 = 121205952119890 (28)




V1 = V2

V2 = minus1198772 [1198861 (V1 minus 120595119889) + 1198862 V2119877 ] (30)









1198911 = 1003816100381610038161003816120595119890Δ120572119903 cos1206011003816100381610038161003816 + (12) 12057421 (Δ120572119903 cos120601)2100381610038161003816100381612057711990310038161003816100381610038162 (34)



1198813 = 1198812 + 121205772119903 (35)



(36)

Note that in (36)

120595119890120576 le 121205952119890 + 1212057621198721198881120595119890120577119903 le 12119888211205952119890 + 121205772119903

1205741120577119903Δ120572119903 cos120601 le 1212057421 (Δ120572119903 cos120601)2 + 121205772119903 (37)

Then we have


(38)


1198814 = 12 log 1198962119903119890

(119905)1198962119903119890

(119905) minus 1199032119890 (39)

where 119896119903119890


119890



1198814 = 12 log 11 minus 1205852119903 (40)


4 = 120599119903( 119903119890 minus 119903119890 119903119890119896119903119890

) (41)

with 120599119903 = 120585119903119896119903119890


4 = 120599119903(119891119903 + 120599T11990311205931199031 + 120599T11990321205931199032 + 120591119903 + 119889119903 minus 119903 minus 119903119890 119903119890119896119903119890

) (42)



119890


where 1198965 = radic(119903119890

119896119903119890



(44)


follows

1198815 = 1198814 + 1212059911987911990311205991199031 + 1212059911987911990321205991199032 + 121198632119903 (45)



119890119896119903119890


(46)


Note that

1198965 + 119903119890119896119903119890

ge 10038161003816100381610038161003816119896510038161003816100381610038161003816 minus 1003816100381610038161003816100381610038161003816100381610038161003816 1199031198901198961199031198901003816100381610038161003816100381610038161003816100381610038161003816 gt 0 (47)


(48)

and

120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) le 02785120590119903119863119903 (49)


5 le minus211989651198814 minus (1198965 + 119903119890119896119903119890




120583119903 = 11989612059912 1003817100381710038171003817120599119903110038171003817100381710038172 + 11989612059922 1003817100381710038171003817120599119903210038171003817100381710038172 + 11989662 1198632119903 + 02785120590119903119863119903 (51)


119890


1198816 = 12 log 11989621199061198901198962119906

119890

minus 1199062119890 (53)


where 120599119906 = 119906119890(1198962119906119890


follows


minus 1199062119890) sin (120595 minus 120574119901) (55)




1198817 = 1198816 + 12120599T11990611205991199061 + 12120599T11990621205991199062 + 121198632119906 (57)



119890


(58)



(59)


120583119906 = 11989612059932 1003817100381710038171003817120599119906110038171003817100381710038172 + 11989612059942 1003817100381710038171003817120599119906210038171003817100381710038172 + 11989682 1198632119906 + 02578120590119906119863119906 (60)



ℎ (119905 119905119886 119905119887 120574119905) = 119891 (120591)119891 (120591) + 120574119905119891 (1 minus 120591) (62)



d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu






119890





(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872






1198901198962119906119890






4 Simulation










4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018









xE

OE yE

zE

(sway)

yb

u((surge)p(roll)

xbr(yaw)

zb

Ob



21 Preliminaries




0 le 10038161003816100381610038161205781003816100381610038161003816 minus 120578 tanh(120578120575) le 02785120575 (1)






= V119903 + (119865119909119886 + 119865119909ℎ + 119865119909119899 + 119865119909119879)119898V = minus119906119903 + (119865119910119886 + 119865119910ℎ + 119865119910119888 + 119865119910119899)119898119903 = (119872119911119886 +119872119911ℎ +119872119911119888 +119872119911119899 +119872119911120575)119868119911

= (119872119909119886 +119872119909ℎ +119872119909119888 +119872119909119899 +119872119909119866)119868119909

(3)


119865119909119886 = 051205881198861198812119886 119878119909119886 (1198621199091198860 + 119862120575119903119909119886120575119903 + 119862120573119909119886120573 + 119862 120573119909119886

120573)119865119910119886 = 051205881198861198812119886 119878119910119886 (1198621199101198860 + 119862120575119903119910119886120575119903 + 119862120573119910119886120573 + 119862 120573

119910119886120573)

119872119911119886 = 051205881198861198812119886 119878119911119886119897119886 (1198621198981199111198860 + 119862120575119903119898119911119886120575119903 + 119862120573119898119911119886120573 + 119862 120573119898119911119886

120573)119872119909119886 = 119865119910119886119911119886

(4)



(5)







(6)



(7)



120574119901 (120596) = atan2 (1199101015840119901 (120596) 1199091015840119901 (120596)) (8)


North

East Xp

pxp

s

yp

e

u

U

X




T [119909 minus 119909119901119910 minus 119910119901] (9)


[[[[[[

120601]]]]]]= [[[[[[


]]]]]][[[[[[

119906V119903119901]]]]]]

(10)





]]]]]+ [[




(12)






Suppose that 119896119904 119896119890 and 119896119906119890


119890


119890




119890









1 = 120599119904 120585119904 + 120599119890 120585119890 (15)




(16)




with



120595119890 = 120595 minus 120595119889 (20)



(21)











(24)

where


sin120595119890120595119890 (25)



119890997888rarr0(sin120595119890120595119890) = 1




where 119896120572119903


where 119896119903119890


Magnitude limiter

rckr

(t)

-kr(t)

+ +

- -

2n

2nn2nn

1

s

1

s

r







119903


119903



1198812 = 121205952119890 (28)




V1 = V2

V2 = minus1198772 [1198861 (V1 minus 120595119889) + 1198862 V2119877 ] (30)









1198911 = 1003816100381610038161003816120595119890Δ120572119903 cos1206011003816100381610038161003816 + (12) 12057421 (Δ120572119903 cos120601)2100381610038161003816100381612057711990310038161003816100381610038162 (34)



1198813 = 1198812 + 121205772119903 (35)



(36)

Note that in (36)

120595119890120576 le 121205952119890 + 1212057621198721198881120595119890120577119903 le 12119888211205952119890 + 121205772119903

1205741120577119903Δ120572119903 cos120601 le 1212057421 (Δ120572119903 cos120601)2 + 121205772119903 (37)

Then we have


(38)


1198814 = 12 log 1198962119903119890

(119905)1198962119903119890

(119905) minus 1199032119890 (39)

where 119896119903119890


119890



1198814 = 12 log 11 minus 1205852119903 (40)


4 = 120599119903( 119903119890 minus 119903119890 119903119890119896119903119890

) (41)

with 120599119903 = 120585119903119896119903119890


4 = 120599119903(119891119903 + 120599T11990311205931199031 + 120599T11990321205931199032 + 120591119903 + 119889119903 minus 119903 minus 119903119890 119903119890119896119903119890

) (42)



119890


where 1198965 = radic(119903119890

119896119903119890



(44)


follows

1198815 = 1198814 + 1212059911987911990311205991199031 + 1212059911987911990321205991199032 + 121198632119903 (45)



119890119896119903119890


(46)


Note that

1198965 + 119903119890119896119903119890

ge 10038161003816100381610038161003816119896510038161003816100381610038161003816 minus 1003816100381610038161003816100381610038161003816100381610038161003816 1199031198901198961199031198901003816100381610038161003816100381610038161003816100381610038161003816 gt 0 (47)


(48)

and

120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) le 02785120590119903119863119903 (49)


5 le minus211989651198814 minus (1198965 + 119903119890119896119903119890




120583119903 = 11989612059912 1003817100381710038171003817120599119903110038171003817100381710038172 + 11989612059922 1003817100381710038171003817120599119903210038171003817100381710038172 + 11989662 1198632119903 + 02785120590119903119863119903 (51)


119890


1198816 = 12 log 11989621199061198901198962119906

119890

minus 1199062119890 (53)


where 120599119906 = 119906119890(1198962119906119890


follows


minus 1199062119890) sin (120595 minus 120574119901) (55)




1198817 = 1198816 + 12120599T11990611205991199061 + 12120599T11990621205991199062 + 121198632119906 (57)



119890


(58)



(59)


120583119906 = 11989612059932 1003817100381710038171003817120599119906110038171003817100381710038172 + 11989612059942 1003817100381710038171003817120599119906210038171003817100381710038172 + 11989682 1198632119906 + 02578120590119906119863119906 (60)



ℎ (119905 119905119886 119905119887 120574119905) = 119891 (120591)119891 (120591) + 120574119905119891 (1 minus 120591) (62)



d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu






119890





(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872






1198901198962119906119890






4 Simulation










4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018












(6)



(7)



120574119901 (120596) = atan2 (1199101015840119901 (120596) 1199091015840119901 (120596)) (8)


North

East Xp

pxp

s

yp

e

u

U

X




T [119909 minus 119909119901119910 minus 119910119901] (9)


[[[[[[

120601]]]]]]= [[[[[[


]]]]]][[[[[[

119906V119903119901]]]]]]

(10)





]]]]]+ [[




(12)






Suppose that 119896119904 119896119890 and 119896119906119890


119890


119890




119890









1 = 120599119904 120585119904 + 120599119890 120585119890 (15)




(16)




with



120595119890 = 120595 minus 120595119889 (20)



(21)











(24)

where


sin120595119890120595119890 (25)



119890997888rarr0(sin120595119890120595119890) = 1




where 119896120572119903


where 119896119903119890


Magnitude limiter

rckr

(t)

-kr(t)

+ +

- -

2n

2nn2nn

1

s

1

s

r







119903


119903



1198812 = 121205952119890 (28)




V1 = V2

V2 = minus1198772 [1198861 (V1 minus 120595119889) + 1198862 V2119877 ] (30)









1198911 = 1003816100381610038161003816120595119890Δ120572119903 cos1206011003816100381610038161003816 + (12) 12057421 (Δ120572119903 cos120601)2100381610038161003816100381612057711990310038161003816100381610038162 (34)



1198813 = 1198812 + 121205772119903 (35)



(36)

Note that in (36)

120595119890120576 le 121205952119890 + 1212057621198721198881120595119890120577119903 le 12119888211205952119890 + 121205772119903

1205741120577119903Δ120572119903 cos120601 le 1212057421 (Δ120572119903 cos120601)2 + 121205772119903 (37)

Then we have


(38)


1198814 = 12 log 1198962119903119890

(119905)1198962119903119890

(119905) minus 1199032119890 (39)

where 119896119903119890


119890



1198814 = 12 log 11 minus 1205852119903 (40)


4 = 120599119903( 119903119890 minus 119903119890 119903119890119896119903119890

) (41)

with 120599119903 = 120585119903119896119903119890


4 = 120599119903(119891119903 + 120599T11990311205931199031 + 120599T11990321205931199032 + 120591119903 + 119889119903 minus 119903 minus 119903119890 119903119890119896119903119890

) (42)



119890


where 1198965 = radic(119903119890

119896119903119890



(44)


follows

1198815 = 1198814 + 1212059911987911990311205991199031 + 1212059911987911990321205991199032 + 121198632119903 (45)



119890119896119903119890


(46)


Note that

1198965 + 119903119890119896119903119890

ge 10038161003816100381610038161003816119896510038161003816100381610038161003816 minus 1003816100381610038161003816100381610038161003816100381610038161003816 1199031198901198961199031198901003816100381610038161003816100381610038161003816100381610038161003816 gt 0 (47)


(48)

and

120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) le 02785120590119903119863119903 (49)


5 le minus211989651198814 minus (1198965 + 119903119890119896119903119890




120583119903 = 11989612059912 1003817100381710038171003817120599119903110038171003817100381710038172 + 11989612059922 1003817100381710038171003817120599119903210038171003817100381710038172 + 11989662 1198632119903 + 02785120590119903119863119903 (51)


119890


1198816 = 12 log 11989621199061198901198962119906

119890

minus 1199062119890 (53)


where 120599119906 = 119906119890(1198962119906119890


follows


minus 1199062119890) sin (120595 minus 120574119901) (55)




1198817 = 1198816 + 12120599T11990611205991199061 + 12120599T11990621205991199062 + 121198632119906 (57)



119890


(58)



(59)


120583119906 = 11989612059932 1003817100381710038171003817120599119906110038171003817100381710038172 + 11989612059942 1003817100381710038171003817120599119906210038171003817100381710038172 + 11989682 1198632119906 + 02578120590119906119863119906 (60)



ℎ (119905 119905119886 119905119887 120574119905) = 119891 (120591)119891 (120591) + 120574119905119891 (1 minus 120591) (62)



d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu






119890





(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872






1198901198962119906119890






4 Simulation










4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018










Suppose that 119896119904 119896119890 and 119896119906119890


119890


119890




119890









1 = 120599119904 120585119904 + 120599119890 120585119890 (15)




(16)




with



120595119890 = 120595 minus 120595119889 (20)



(21)











(24)

where


sin120595119890120595119890 (25)



119890997888rarr0(sin120595119890120595119890) = 1




where 119896120572119903


where 119896119903119890


Magnitude limiter

rckr

(t)

-kr(t)

+ +

- -

2n

2nn2nn

1

s

1

s

r







119903


119903



1198812 = 121205952119890 (28)




V1 = V2

V2 = minus1198772 [1198861 (V1 minus 120595119889) + 1198862 V2119877 ] (30)









1198911 = 1003816100381610038161003816120595119890Δ120572119903 cos1206011003816100381610038161003816 + (12) 12057421 (Δ120572119903 cos120601)2100381610038161003816100381612057711990310038161003816100381610038162 (34)



1198813 = 1198812 + 121205772119903 (35)



(36)

Note that in (36)

120595119890120576 le 121205952119890 + 1212057621198721198881120595119890120577119903 le 12119888211205952119890 + 121205772119903

1205741120577119903Δ120572119903 cos120601 le 1212057421 (Δ120572119903 cos120601)2 + 121205772119903 (37)

Then we have


(38)


1198814 = 12 log 1198962119903119890

(119905)1198962119903119890

(119905) minus 1199032119890 (39)

where 119896119903119890


119890



1198814 = 12 log 11 minus 1205852119903 (40)


4 = 120599119903( 119903119890 minus 119903119890 119903119890119896119903119890

) (41)

with 120599119903 = 120585119903119896119903119890


4 = 120599119903(119891119903 + 120599T11990311205931199031 + 120599T11990321205931199032 + 120591119903 + 119889119903 minus 119903 minus 119903119890 119903119890119896119903119890

) (42)



119890


where 1198965 = radic(119903119890

119896119903119890



(44)


follows

1198815 = 1198814 + 1212059911987911990311205991199031 + 1212059911987911990321205991199032 + 121198632119903 (45)



119890119896119903119890


(46)


Note that

1198965 + 119903119890119896119903119890

ge 10038161003816100381610038161003816119896510038161003816100381610038161003816 minus 1003816100381610038161003816100381610038161003816100381610038161003816 1199031198901198961199031198901003816100381610038161003816100381610038161003816100381610038161003816 gt 0 (47)


(48)

and

120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) le 02785120590119903119863119903 (49)


5 le minus211989651198814 minus (1198965 + 119903119890119896119903119890




120583119903 = 11989612059912 1003817100381710038171003817120599119903110038171003817100381710038172 + 11989612059922 1003817100381710038171003817120599119903210038171003817100381710038172 + 11989662 1198632119903 + 02785120590119903119863119903 (51)


119890


1198816 = 12 log 11989621199061198901198962119906

119890

minus 1199062119890 (53)


where 120599119906 = 119906119890(1198962119906119890


follows


minus 1199062119890) sin (120595 minus 120574119901) (55)




1198817 = 1198816 + 12120599T11990611205991199061 + 12120599T11990621205991199062 + 121198632119906 (57)



119890


(58)



(59)


120583119906 = 11989612059932 1003817100381710038171003817120599119906110038171003817100381710038172 + 11989612059942 1003817100381710038171003817120599119906210038171003817100381710038172 + 11989682 1198632119906 + 02578120590119906119863119906 (60)



ℎ (119905 119905119886 119905119887 120574119905) = 119891 (120591)119891 (120591) + 120574119905119891 (1 minus 120591) (62)



d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu






119890





(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872






1198901198962119906119890






4 Simulation










4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018

















(24)

where


sin120595119890120595119890 (25)



119890997888rarr0(sin120595119890120595119890) = 1




where 119896120572119903


where 119896119903119890


Magnitude limiter

rckr

(t)

-kr(t)

+ +

- -

2n

2nn2nn

1

s

1

s

r







119903


119903



1198812 = 121205952119890 (28)




V1 = V2

V2 = minus1198772 [1198861 (V1 minus 120595119889) + 1198862 V2119877 ] (30)









1198911 = 1003816100381610038161003816120595119890Δ120572119903 cos1206011003816100381610038161003816 + (12) 12057421 (Δ120572119903 cos120601)2100381610038161003816100381612057711990310038161003816100381610038162 (34)



1198813 = 1198812 + 121205772119903 (35)



(36)

Note that in (36)

120595119890120576 le 121205952119890 + 1212057621198721198881120595119890120577119903 le 12119888211205952119890 + 121205772119903

1205741120577119903Δ120572119903 cos120601 le 1212057421 (Δ120572119903 cos120601)2 + 121205772119903 (37)

Then we have


(38)


1198814 = 12 log 1198962119903119890

(119905)1198962119903119890

(119905) minus 1199032119890 (39)

where 119896119903119890


119890



1198814 = 12 log 11 minus 1205852119903 (40)


4 = 120599119903( 119903119890 minus 119903119890 119903119890119896119903119890

) (41)

with 120599119903 = 120585119903119896119903119890


4 = 120599119903(119891119903 + 120599T11990311205931199031 + 120599T11990321205931199032 + 120591119903 + 119889119903 minus 119903 minus 119903119890 119903119890119896119903119890

) (42)



119890


where 1198965 = radic(119903119890

119896119903119890



(44)


follows

1198815 = 1198814 + 1212059911987911990311205991199031 + 1212059911987911990321205991199032 + 121198632119903 (45)



119890119896119903119890


(46)


Note that

1198965 + 119903119890119896119903119890

ge 10038161003816100381610038161003816119896510038161003816100381610038161003816 minus 1003816100381610038161003816100381610038161003816100381610038161003816 1199031198901198961199031198901003816100381610038161003816100381610038161003816100381610038161003816 gt 0 (47)


(48)

and

120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) le 02785120590119903119863119903 (49)


5 le minus211989651198814 minus (1198965 + 119903119890119896119903119890




120583119903 = 11989612059912 1003817100381710038171003817120599119903110038171003817100381710038172 + 11989612059922 1003817100381710038171003817120599119903210038171003817100381710038172 + 11989662 1198632119903 + 02785120590119903119863119903 (51)


119890


1198816 = 12 log 11989621199061198901198962119906

119890

minus 1199062119890 (53)


where 120599119906 = 119906119890(1198962119906119890


follows


minus 1199062119890) sin (120595 minus 120574119901) (55)




1198817 = 1198816 + 12120599T11990611205991199061 + 12120599T11990621205991199062 + 121198632119906 (57)



119890


(58)



(59)


120583119906 = 11989612059932 1003817100381710038171003817120599119906110038171003817100381710038172 + 11989612059942 1003817100381710038171003817120599119906210038171003817100381710038172 + 11989682 1198632119906 + 02578120590119906119863119906 (60)



ℎ (119905 119905119886 119905119887 120574119905) = 119891 (120591)119891 (120591) + 120574119905119891 (1 minus 120591) (62)



d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu






119890





(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872






1198901198962119906119890






4 Simulation










4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018












1198911 = 1003816100381610038161003816120595119890Δ120572119903 cos1206011003816100381610038161003816 + (12) 12057421 (Δ120572119903 cos120601)2100381610038161003816100381612057711990310038161003816100381610038162 (34)



1198813 = 1198812 + 121205772119903 (35)



(36)

Note that in (36)

120595119890120576 le 121205952119890 + 1212057621198721198881120595119890120577119903 le 12119888211205952119890 + 121205772119903

1205741120577119903Δ120572119903 cos120601 le 1212057421 (Δ120572119903 cos120601)2 + 121205772119903 (37)

Then we have


(38)


1198814 = 12 log 1198962119903119890

(119905)1198962119903119890

(119905) minus 1199032119890 (39)

where 119896119903119890


119890



1198814 = 12 log 11 minus 1205852119903 (40)


4 = 120599119903( 119903119890 minus 119903119890 119903119890119896119903119890

) (41)

with 120599119903 = 120585119903119896119903119890


4 = 120599119903(119891119903 + 120599T11990311205931199031 + 120599T11990321205931199032 + 120591119903 + 119889119903 minus 119903 minus 119903119890 119903119890119896119903119890

) (42)



119890


where 1198965 = radic(119903119890

119896119903119890



(44)


follows

1198815 = 1198814 + 1212059911987911990311205991199031 + 1212059911987911990321205991199032 + 121198632119903 (45)



119890119896119903119890


(46)


Note that

1198965 + 119903119890119896119903119890

ge 10038161003816100381610038161003816119896510038161003816100381610038161003816 minus 1003816100381610038161003816100381610038161003816100381610038161003816 1199031198901198961199031198901003816100381610038161003816100381610038161003816100381610038161003816 gt 0 (47)


(48)

and

120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) le 02785120590119903119863119903 (49)


5 le minus211989651198814 minus (1198965 + 119903119890119896119903119890




120583119903 = 11989612059912 1003817100381710038171003817120599119903110038171003817100381710038172 + 11989612059922 1003817100381710038171003817120599119903210038171003817100381710038172 + 11989662 1198632119903 + 02785120590119903119863119903 (51)


119890


1198816 = 12 log 11989621199061198901198962119906

119890

minus 1199062119890 (53)


where 120599119906 = 119906119890(1198962119906119890


follows


minus 1199062119890) sin (120595 minus 120574119901) (55)




1198817 = 1198816 + 12120599T11990611205991199061 + 12120599T11990621205991199062 + 121198632119906 (57)



119890


(58)



(59)


120583119906 = 11989612059932 1003817100381710038171003817120599119906110038171003817100381710038172 + 11989612059942 1003817100381710038171003817120599119906210038171003817100381710038172 + 11989682 1198632119906 + 02578120590119906119863119906 (60)



ℎ (119905 119905119886 119905119887 120574119905) = 119891 (120591)119891 (120591) + 120574119905119891 (1 minus 120591) (62)



d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu






119890





(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872






1198901198962119906119890






4 Simulation










4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018









Note that

1198965 + 119903119890119896119903119890

ge 10038161003816100381610038161003816119896510038161003816100381610038161003816 minus 1003816100381610038161003816100381610038161003816100381610038161003816 1199031198901198961199031198901003816100381610038161003816100381610038161003816100381610038161003816 gt 0 (47)


(48)

and

120599119903 (119889119903 minus 119863119903 tanh(120599119903120590119903)) le 02785120590119903119863119903 (49)


5 le minus211989651198814 minus (1198965 + 119903119890119896119903119890




120583119903 = 11989612059912 1003817100381710038171003817120599119903110038171003817100381710038172 + 11989612059922 1003817100381710038171003817120599119903210038171003817100381710038172 + 11989662 1198632119903 + 02785120590119903119863119903 (51)


119890


1198816 = 12 log 11989621199061198901198962119906

119890

minus 1199062119890 (53)


where 120599119906 = 119906119890(1198962119906119890


follows


minus 1199062119890) sin (120595 minus 120574119901) (55)




1198817 = 1198816 + 12120599T11990611205991199061 + 12120599T11990621205991199062 + 121198632119906 (57)



119890


(58)



(59)


120583119906 = 11989612059932 1003817100381710038171003817120599119906110038171003817100381710038172 + 11989612059942 1003817100381710038171003817120599119906210038171003817100381710038172 + 11989682 1198632119906 + 02578120590119906119863119906 (60)



ℎ (119905 119905119886 119905119887 120574119905) = 119891 (120591)119891 (120591) + 120574119905119891 (1 minus 120591) (62)



d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu






119890





(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872






1198901198962119906119890






4 Simulation










4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018









d1u

d2u

00 at bt

T [s]

and

its d

eriv

ativ

edu

dudu






119890





(64)

where 120589119881 = min1205891 1205892 1205893 1205894 and 120583119881 = 120583119906 + 120583119903 + (12)1205762119872






1198901198962119906119890






4 Simulation










4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018












4

5

u [ms]

r [de

gs]


8 10

3

212 14

45



(69)



119890



119903


119890

(119905) le 119896119903(119905)minus119896120572119903












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018












Case 1Case 2

2600 2700 2800 2900

minus2000

minus1800

20 40 60 800

200

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

1500

2000

2500

y [m

]

0 500 1000 1500 2000 2500 3000 3500minus500x [m]




119903



119890

= 1 degs le 119896119903(119905) minus 119896120572119903

(119905)

0 100 200 300 400 500 600 700 800 900minus50minus30

0

3050

Time [s]

x e [m

]


0 100 200 300 400 500 600 700 800 900minus50

minus30

0

3050

Time [s]

y e [m

]






406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018










406 408 4108

9

10

11

12

100 200 300 400 500 600 700 800 9000Time [s]

6

7

8

9

10

11

12

13

u [m

s]


minus5

0

5

r [de

gs]


minus5

0

5

r [de

gs]

minus5

0

5

r [de

gs]



100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]



0 200 400 600 800minus3

minus2

minus1

0

1

2

3

Time [s]

r e [d

egs

]


0 10 20 30minus1

0

1




119890


5 Conclusion



x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018









x 104

x 105

x 104

x 104

minus8

minus6

minus4

1

100 200 300 400 500 600 700 800 9000Time [s]

minus5

0

5

2

minus2

minus1

0

3

minus2

0

2

4

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]


＄Ｏ＞Ｏ

＄Ｌ

＞Ｌ

minus002

0

002

004

＄Ｌ

100 200 300 400 500 600 700 800 9000Time [s]

100 200 300 400 500 600 700 800 9000Time [s]

minus01

0

01

02

＄Ｏ



0 200 400 600 800Time [s]


200 400 600 8000Time [s]

minus04

minus02

0

02

04

Ｌ

minus1

0

1

2



Appendix






119898

minus 120588119886119876119888 (2119901119888120588119886 )12 119897119866

119888

) 119891119901 = 1119868119909 (minus120588119886119876119888 (2119901119888120588119886 )



1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018









1205991199061 = 119878119909119886 [1198621199091198860 119862120575119903119909119886 119862120573119909119886 119862 120573119909119886]T


119910119886]T 120593V1 = 1205931199061120599V2 = 119878ℎ [1198620119910ℎ 119862120601119910ℎ 119862120573119910ℎ]T 120593V2 = 12059311990621205991199031 = 119878119911119886 [1198621198981199111198860 119862120575119903119898119911119886 119862120573119898119911119886 119862 120573

119898119911119886]T

1205931199031 = 119902119886119897119886119868119911 [1 120575119903 120573 120573]T 1205991199032 = 119878ℎ [119862120601119898119911ℎ 119862120573119898119911ℎ]T1205931199032 = 119902ℎ119897119888119868119911 [120601 120573]T 1205991199011 = 120599V11205991199012 = 120599V21205931199011 = 119902119886119911119886119868119909 [1 120575119903 120573 120573]T 1205931199012 = 119902ℎ119911119892119868119909 [120601 120573]T

(B1)

Data Availability




Acknowledgments


References































































Journal of












Journal of







Volume 2018






Volume 2018




















































Journal of












Journal of







Volume 2018






Volume 2018








barrier lyapunov function-based adaptive control of an

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