Global robust adaptive path following of underactuated ships

Automatica42(2006)1713–1722/locate/automaticaBrief paperGlobal robust adaptive path following of underactuated shipsଁK.D.Do∗,J.PanSchool of Mechanical Engineering,The University of Western Australia,Crawley,WA6009,AustraliaReceived11April2005;received in revised form24April2006;accepted30April2006Available online7July2006AbstractWe propose a method for designing a global robust adaptive controller that forces an underactuated ship to follow a reference path under both constant and time-varying disturbances induced by waves,wind and ocean-currents.Both linear and nonlinear damping terms are included to cover both low-and high-speed applications.All nonlinear damping coefficients are assumed unknown but lie in a known compact set. The new results are derived using a choice of an appropriate body-fixed frame origin,a smooth approximation of nonsmooth damping terms, several nonlinear coordinate changes,the backstepping technique,and utilization of the ship dynamic structure.Experiments on a model ship illustrate the results.᭧2006Elsevier Ltd.All rights reserved.Keywords:Underactuated ships;Path following;Adaptive and robust control1.IntroductionSteering a ship along a desired path with prescribed forward speed is an important issue in many offshore applications.This goal can be achieved by solving trajectory-tracking and/or path-following problems(Fossen,2002).The main difficulty with both trajectory-tracking and path-following issues is that the sway axis is not actuated.This configuration is by far most com-mon among the marine surface vessels(Fossen,2002).Hence, the control problem of underactuated ships has received a lot of attention from the control community.Encarnacao,Pacoal,and Arcak(2000)described the path following errors in the Serret-Frenet frame,then designed a local path following controller under constant ocean-current.Pettersen and Nijmeijer(2001) provided a high-gain,local exponential tracking result.By ap-plying a cascade approach,a global tracking result was obtained in Lefeber,Pettersen,and Nijmeijer(2003).Two tracking so-lutions were proposed in Jiang(2002)using Lyapunov’s direct and passivity approaches.A high-gain based global practical ଁThis paper was not presented at any IFAC meeting.This paper was recommended for publication in revised form by Associate Editor Thor I. Fossen under the direction of Editor Hassan Khalil.∗Corresponding author.Tel.:+61864883602;fax:+6164881024.E-mail addresses:duc@.au(K.D.Do),pan@.au(J.Pan).0005-1098/$-see front matter᭧2006Elsevier Ltd.All rights reserved. doi:10.1016/j.automatica.2006.04.026tracking controller was developed in Behal,Dawson,Dixon, and Fang(2002)using a transformation of the ship tracking system into a skew-symmetric form.It is noted that in Jiang (2002),Pettersen and Nijmeijer(2001)and Lefeber et al.(2003), the yaw speed was required to be nonzero.This restrictive as-sumption was removed in Do,Jiang,and Pan(2002).It seems that thefirst global way-point tracking controller was proposed in Pettersen and Lefeber(2001)to force an underactuated ship to track a straight-line,see also Fredriksen and Pettersen(2004) for an improvement.Relevant independent work also includes Sira-Ramirez(1999)on differentialflatness approach.Based on a line-of-sight projection algorithm,Fossen,Breivik,and Skjetne(2003)proposed a controller to force an underactuated ship to follow a sequence of straight-line segments connected by way-points.It is noted that in Pettersen and Nijmeijer(2001),Lefeber et al.(2003),Jiang(2002),Do,Jiang,and Pan(2002),and Sira-Ramirez(1999),the mass and damping matrices of the ships are assumed to be diagonal.Relaxing these assumptions destroys the cascade structure,which is crucial for control design and stability analysis.The nonlinear damping terms are also ignored,i.e.high-speed applications are excluded.Fur-thermore,the ships usually operate in open sea subject to en-vironmental disturbances.Therefore,the control systems must consider these disturbances.Recently,we obtained a solution1714K.D.Do,J.Pan/Automatica42(2006)1713–1722 that removed the above assumptions but did not include thenonlinear damping terms(Do&Pan,2005).However whennonlinear damping terms are included in the model,we facethree difficult problems:(1)all nonlinear damping coefficientsare difficult to obtain accurately by semi-empirical methods orhydrodynamic programs;(2)nonlinear damping terms intro-duce nonsmooth terms to the model,which create a difficultywhen applying the backstepping technique;and(3)stabilityof the sway dynamics is difficult to analyze.In this paper,wepropose an approach that removes all of the aforementionedassumptions.In this approach,the derivative of the path pa-rameter is used as an additional control.The new approach isbased on a choice of an appropriate body-fixed frame origin,a smooth approximation of nonsmooth damping terms,severalnonlinear coordinate changes,thebackstepping technique,andutilization of the ship dynamic structure.2.Problem formulationAssume that the ship has an xz-plane of symmetry;heave,pitch and roll modes are neglected;the body-fixed frame co-ordinate origin is set in the center-line of the ship.Then themathematical model of an underactuated ship moving in a hor-izontal plane is described as(Fossen,2002):˙ =J( )v,M˙v=−C(v)v−(D+D n(v))v+ +J T( ) c+ w(t)(1)with=[x y ]T,v=[u v r]T, =[ u0 r]T,c=[ cu cv cr]T, w(t)=[ wu(t) wv(t) wr(t)]T,J( )= cos( )−sin( )0sin( )cos( )0001,M=m11000m22m230m32m33,C= 00C1300C23C31C320,D+D n(v)=D11000D22D230D32D33,m11=m−X˙u,m22=m−Y˙v,m23=mx g−Y˙r,m32=mx g−N˙v,m33=I z−N˙r,C13=−C31=−m22v−m22s r,C23=−C32=m11u,D11=−(X u+X|u|u|u|),D22=−(Y v+Y|v|v|v|+Y|r|v|r|), D23=−(Y r+Y|v|r|v|),D32=−(N v+N|v|v|v|+N|r|v|r|), D33=−(N r+N|v|r|v|+N|r|r|r|),where m22s=0.5(m23+m32),(x,y, )denote the position and orientation of the ship with coordinates in the earth-fixed frame; u,v and r denote surge,sway and yaw speeds with coordinates in the body-fixed frame;m is the mass of the ship;I z is theYydy dFig.1.Interpretation of path following errors.ship’s inertia about the Z b-axis of the body-fixed frame;x g is the X b-coordinate of the ship center of gravity in the body-fixed frame(see Fig.1);the controls u and r are the surge force and yaw moment in the body-fixed frame; c and w(t)are constant and time-varying disturbances.Notice that we use w(t)in the body-fixed frame to shorten later expressions.Treating w(t)in the earth-fixed frame isthe same but only the magnitude of w(t) is different.The other symbols are referred to as hydrodynamic derivatives(SNAME,1950).Control objective:Under Assumptions1and2,design the controls u and r to force the ship(1)to follow a prescribed path parameterized by(x d(s),y d(s))with s being the path parameter in the sense that the position of the ship(1)tracks the path ,the ship total velocity is tangential to the path , and let the desired surge speed,u0(t),be adjustable.Assumption1.(1)The ship parameters,added masses and lin-ear damping coefficients:m,I z,X˙u,Y˙v,Y˙r,N˙v,X u,Y v,Y r,N v, N r are known.(2)The nonlinear damping coefficients:X|u|u,Y|v|v,Y|r|v, Y|v|r,N|v|v,N|r|v,N|v|r,N|r|r are unknown but lie in a known compact set,i.e.there exist known positive constants X|u|u, Y|v|v, Y|r|v, Y|v|r, N|v|v, N|r|v, N|v|r, N|r|r such that|X|u|u| X|u|u,|Y|v|v| Y|v|v,|Y|r|v| Y|r|v,|Y|v|r| Y|v|r,|N|v|v| N|v|v,|N|r|v| N|r|v,|N|v|r| N|v|r,|N|r|r| N|r|r. (3)The disturbance forces and moments are unknown but lie in a known compact set,i.e.there exist known positive constants M cu, M cv, M cr, M wu, M wv, M wr such that| cu| M cu,| cv| M cv,| cr| M cr,| wu(t)| M wu,| wv(t)| M wv,| wr(t)| M wr,∀t∈R+.(4)There exist strictly positive constants¯u∗d,u∗0,u∗01,u∗02and u M0such thatx d2(s)+y d2(s) ¯u∗d,∀s∈R,u∗0 u0(t) u M0,|˙u0(t)| u∗01,|¨u0(t)| u∗02,∀t∈R+.(2)K.D.Do,J.Pan /Automatica 42(2006)1713–17221715Remark 1.Several software packages (such as VERES from Marintek)can be used to calculate all the ship parameters,added masses and linear damping coefficients accurately.How-ever,the coefficients of nonlinear damping terms are difficult to obtain accurately.Therefore Parts (1)and (2)of Assumption 1are reasonable.Part (4)implies that the path is regular with respect to the path parameter s .The desired surge speed is always strictly positive,i.e.we do not consider stabiliza-tion/regulation problems.Assumption 2.There exist strictly positive constants k 1,k 2,|v |r , |r |v ,b ∗i , i,i =1,2,3such that one of the following sets of conditions holds:(1)Set I :1−m 11m −122−[0.65m −122(( Y|v |r (1+ |v |r )+ Y |r |v (1+ |v |r ))]2−{k 2+m −122×(k 1m 11+1.3k 2[(( Y |v |r (1+ |v |r )+ Y |r |v (1+ |v |r )]−Y r )}/u ∗0 b ∗1,Y |r |v +|Y |v |r | − 1.(2)Set II :1−m 11m −122−0.65m −122(0.5+k 2/u ∗0)[( Y |v |r (1+ |v |r )+ Y |r |v (1+ |v |r )]−(k 2+k 1m 11m −122−Y r m −122)/u ∗0 b ∗2,Y |v |v +(Y |r |v −|Y |v |r |)|¯r d (s)|/(b ∗2¯u d (s)) − 2,∀s ∈R .(3)Set III :X |u |u =Y |v |v =Y |r |v =Y |v |r =N |v |v =N |r |v =N |v |r=N |r |r =0,1−m 11m −122−(k 2+m 11k 1m −122−Y r m −122)/u ∗0 b ∗3,Y v +(m 11(k 1+u M 0)+|Y r |)|¯r d (s)|/(b ∗2¯u d (s)) − 3,s ∈RWe have defined ¯r d (s)and ¯u d (s)as¯u d (s)=x d 2(s)+y d2(s),¯r d (s)=(x d (s)y d (s)−x d (s)y d(s))¯u 2d (s),(3)where (•) =j (•)/j s and (•) =j 2(•)/j s 2.From nowonward,we drop the argument s of ¯r d (s)¯u d (s).Remark 2.The first conditions of Sets I and II,and the second condition of Set III are needed so that the control problem is solvable.The second conditions of Sets I and II,and the third condition of Set III are needed to guarantee boundedness of the sway velocity.The first conditions of Sets I and II imply that:the control design parameters (k 1,k 2, |v |r , |r |v )cannot be arbitrarily large,u ∗0should not be too small,|Y |v |r |and |Y |r |v |are not too large compared with m 22,and m 11<m 22.The main differences between Sets I and II are:Set I allows bigger value of |Y |v |r |and |Y |r |v |,and arbitrarily large path curvaturebut requires Y |r |v <|Y |r |v |.On the other hand Set II does notrequire Y |r |v <|Y |v |r |but cannot allow the path curvature (viathe ratio ¯r d /¯ud )to be arbitrarily large.Set III implies that when the nonlinear damping terms are neglected,the curvature of the desired path is restricted by Y v and Y r .The main ideas to solve the control objective are:1.Choose an appropriate body-fixed frame origin to avoid the yaw moment r acting directly on the sway dynamics.2.Interpret path-following errors in a frame attached to the path such that the error dynamics are of a triangular form to which the backstepping technique can be applied.e the orientation error as a virtual control to stabilize the cross-track error where the derivative of the path parameter is used as a control to design a global controller.e smooth approximations of |r |and |v |when applying the backstepping technique to avoid the use of nonsmooth control design techniques.3.Coordinate transformations 3.1.Choosing body-fixed frame originTo avoid the yaw moment r acting directly on the sway dynamics,we choose the body-fixed frame origin such that it is on the center-line of the ship (see Fig.1),i.e.y g =0and that y g =Y ˙r /m .It is noted that although this choice of the body-fixed frame origin is somewhat uncertain due to the uncertainty of Y ˙r ,our experimental results show that the proposed controller can handle this uncertainty very well.As such,the model (1)becomes˙x =u cos ( )−v sin ( ),˙y =u sin ( )+v cos ( ),˙=r ,˙v =m −122(−m 11ur +(Y r +Y |v |r |v |)r +(Y |r |v |r |+Y v +Y |v |v |v |)v−sin ( ) cu +cos ( ) cv + wv (t)),˙u =m −111[(m 22v +m 22s r)r +(X u +X |u |u |u |)u + u+cos ( ) cu +sin ( ) cv + wu (t)],˙r =m 32m −122m −133[m 11ur −(Y r +Y |v |r |v |)r −(Y |r |v |r |+Y v +Y |v |v |v |)v +sin ( ) cu −cos ( ) cv− wv (t)]+m −133[−(m 22v +m 22s r)u +m 11uv +(N v +N |v |v |v |+N |r |v |r |)v +(N r +N |v |r |v |+N |r |r |r |)r + r + cr + wr (t)].(4)3.2.Transforming path-following errorsWe now interpret the path-following errors in a frame at-tached to the path (Samson,1991)as (see Fig.1):[x ey ee ]T =J T ( )[x −x d y −y d − d ]T(5)1716K.D.Do,J.Pan/Automatica42(2006)1713–1722 where d is the angle between the path and the X-axis definedby d=arctan(y d(s)/x d(s))with x d(s)and y d(s)defined in(3).In Fig.1,OXY is the earth-fixed frame;O p X p Y p is a frameattached to the path such that O p X p and O p Y p are parallelto the surge and sway axes of the ship;u d is tangential to thepath;O c is the center of gravity of the ship;and O b X b Y b isthe body-fixed frame.Therefore x e,y e and e are referred to astangential,cross and heading errors,respectively.Differentiat-ing both sides of along the solutions of thefirst three equationsof results in the kinematic error dynamics:˙x e=u−u d cos( e)+ry e,˙y e=v+u d sin( e)−rx e,˙e=r−r d,(6)where u d and r d are given byu d=¯u d˙s,r d=¯r d˙s(7)with¯u d and¯r d given in(3).From(6),it can be seen that(x e,y e, e)=(0,0,0)is the equilibrium point if only the swayvelocity is zero,i.e.v=0,which means that the ship must moveon a straight-line at the steady state.Since we allow the pathto be different from a straight-line,but also include it,the swayvelocity is generally different from zero at the steady state.InDo and Pan(2005),we resolved this difficulty by introducingan angle to the orientation error ing that approach,wefind it impossible to include the nonlinear damping terms.Thispaper presents a different approach to design u and r in thenext section.4.Control designFor convenience of the reader,we divide the control designprocedure into two separate stages.Firstly,we consider(6)andthefifth equation of(4).Secondly,the last two equations of(4)are considered to design the controls u and r using thebackstepping technique found in Khalil(2002).4.1.Kinematic control designSub-step1.1.Stabilizing(x e,y e)-dynamics:Defineu e=u− u,¯ e= e−e,(8)where u ande are virtual controls of u and e.Substituting(8)into thefirst two equations of(6)gives˙x e= u+u e−u d cos(e )+ry e+ 1,˙y e=v+u d sin(e)−rx e+ 2,(9) where1=−u d((cos(¯ e)−1)cos(e )−sin(¯ e)sin(e)),2=u d(sin(¯ e)cos(e )+(cos(¯ e)−1)sin(e)).(10)From thefirst equation of(9),we design a control law for uto stabilize the x e-dynamics asu=− 1+u d cos(e), 1=k1x e/ , =1+x2e+y2e,(11)where k1is a positive constant,which is chosen as in As-sumption2.Stabilizing the second equation of(9)is more in-volved.We cannot directly useeto cancel v since the termu d sin(e)cannot cancel v without restriction on the initialconditions and on the size of the disturbances in the sway dy-namics.The idea to overcome this difficulty is to design u d insuch a way that it guarantees the desired surge velocity at thesteady state,and dominates v.Hence,we use the derivative ofthe path parameter as a control designed as˙s=u20+( 2+v)2/¯u d, 2=k2y e/ ,(12)where k2is a positive constant chosen as in Assumption2.With(12),we haveu d=u20+( 2+v)2.(13)From(12),the second equation of(9),and(7),we design acontrol law forease=−arctan(( 2+v)/u0).(14)Substituting(14),(12)and(11)and into(9)results in˙x e=− 1+u e+ry e+ 1,˙y e=− 2−rx e+ 2.(15)On the other hand,substituting(14)into(11)results inu=− 1+u0.(16)Remark3.At the steady state,with the help of a control(tobe designed later),convergence of(x e,y e,¯ e,u e)to an ar-bitrarily small value implies that e converges to0.5( ,− )plus/minus an arbitrarily small value.This implies that the shipwill not turn around.From(16),the surge velocity u(t)willapproach u0plus/minus an arbitrarily small value,and u d(t)will approach the total linear speed:u20+v(t)2plus/minus anarbitrarily small value.Sub-step1.2.Stabilizing¯ e dynamics:Differentiating bothsides of the second equation of(8)along the solutions of(15)and the fourth equation of(4)results in˙¯e=r−r d−˙u0( 2+v)u d−k2u0x eu dr+k2u0u2d 3[(1+x2e)(− 2+ 2)−x e y e(− 1+ 1)]−k2u0x e y e u eu2d 3+u0m22u2d[−m11ur+(Y r+Y|v|r|v|)×r+(Y|r|v|r|+Y v+Y|v|v|v|)v−sin( ) cu+cos( ) cv+ wv(t)].(17)K.D.Do,J.Pan /Automatica 42(2006)1713–17221717From (17),if r is considered as a control to stabilize the ¯ edynamics,we face a nonsmooth problem due to |r |and |v |.The nonsmooth control design technique in Tanner and Kyriakopoulos (2003)can be modified and applied to our case,but computing Filippov’s differential inclu-sion and nonsmooth stability analysis are required.We here use a simple way to get around the problem.Adding andsubtracting the terms u 0m −122u −2d ¯Y |v |r vr tanh (¯ e vr/( 1 3)),u 0m −122u −2d ¯Y |r |v vr tanh (¯ e vr/( 2 3)),u 0m −122u −2d ¯Y |v |v v 2tanh (¯ e v 2/( 3 3)), M wv m −122u −2d tanh (¯ e /( 4 3)),where ¯Y|v |r =|Y |v |r |,¯Y |r |v =|Y |r |v |,¯Y |v |v =|Y |v |v |,and i ,1 i 4are small positive constants to be specified later,to the right hand side of (17)result in˙¯ e =b 1−r d +u 0m −122u −2d[¯Y |v |v v 2tanh (¯ e v 2/( 3 3))−sin ( ) cu +cos ( ) cv + M wvtanh (¯e/( 4 3))+˜¯Y |v |r vr tanh (¯ evr/( 1 3))+˜¯Y |r |v vr tanh (¯ evr/( 2 3))]+b 2u e + 1+ 3(18)where b 1=r −k 2u 0x e u dr −u 0m 22u d [m 11 u r +Y r r +ˆ¯Y |v |r vr tanh (¯ evr/( 1 3))+ˆ¯Y|r |vvr tanh (¯ evr/( 23))],b 2=−k 2u 0x e y e u −2d −3−m 11u 0rm −122u −2d , 3=−˙u 0( 2+v)u −2d+k 2u 0u −2d −3((1+x 2e )(− 2+ 2)−x e y e (− 1+ 1))+Y v u 0vm −122u −2d ,1=u 0m −122u −2d [Y |v |r |v |r −¯Y |v |r vr tanh (¯ e vr/( 1 3))+Y |r |v |r |v −¯Y |r |v vr tanh (¯ evr/( 2 3))+Y |v |v |v |v −¯Y|v |v v 2tanh (¯ ev 2/( 3 3))+ wv (t)− M wv tanh (¯ e /( 4 3))](19)where (ˆ•)denotes an estimate of (•),and (˜•)=(•)−(ˆ•).Theˆ¯Y|v |r and ˆ¯Y |r |v estimates are updated as follows:˙ˆ¯Y |v |r = 1proj (¯ e vr tanh (¯ evr/( 1 3)),ˆ¯Y |v |r ),˙ˆ¯Y |r |v = 2proj (¯ e vr tanh (¯ evr/( 2 3)),ˆ¯Y |r |v ),(20)where 1and 2are positive constants to be specified later.The operator proj represents the Lipschitz projection algorithm (Pomet &Praly,1992)asproj ( ,ˆ )= if (ˆ ) 0,proj ( ,ˆ )= if (ˆ ) 0and ˆ (ˆ ) 0,proj ( ,ˆ )=(1− (ˆ )) if (ˆ )>0and ˆ (ˆ) >0,where (ˆ )=(ˆ 2− 2M )/( 2+2 M ), ˆ(ˆ )=j (ˆ )/j ˆ , is an arbitrarily small positive constant (ASPC),and | | M .The projection algorithm is such that if ˙ˆ =proj ( ,ˆ )andˆ (t 0) M then (a)ˆ (t) M + ,∀0 t 0 t <∞;(b)proj ( ,ˆ )is Lipschitz continuous;(c)|proj ( ,ˆ )| | |;(d)˜ proj ( ,ˆ ) ˜ with ˜ = −ˆ .The reason we use the above projection algorithm is to guar-antee boundedness of ˆ¯Y|r |v and ˆ¯Y |v |r ,i.e.ˆ¯Y |v |r Y |v |r (1+ |v |r ),ˆ¯Y |r |v Y |r |v (1+ |r |v ),(21)where |v |r and |r |v are ASPCs.From (18),we consider b 1asa control to stabilize the ¯ e dynamics.Define r e =b 1− r ,(22)where r is a virtual control of b 1.We substitute (22)into (18)and choose the virtual control r asr =−k 3¯ e +r d −u 0m −122u −2d[ˆ¯Y |v |v v 2tanh (¯ e v 2/( 3 3))−sin ( )ˆ cu +cos ( )ˆcv + M wvtanh (¯ e/( 4 3))]− 3,(23)where k 3is a positive constant.The estimates ˆ cu ,ˆ cv will beupdated later.The estimate ˆ¯Y|v |v is updated as follows:˙ˆ¯Y |v |v = 3proj (¯ e v 2tanh (¯ ev 2/( 3 3)),ˆ¯Y |v |v ),(24)where 3is a positive constant to be specified later.It is notedthat (24)guarantees that ˆ¯Y |v |v Y |v |v (1+ |v |v ),(25)where |v |v is an ASPC.Substituting (22)and (23)into (18)results in˙¯ e =−k 3¯ e+r e +b 2u e +u 0m −122u −2d[˜¯Y |v |r vr tanh (¯ evr/( 1 3))+˜¯Y |r |v vr tanh (¯ evr/( 2 3))+˜¯Y|v |vv 2tanh (¯ ev 2/( 33)]−sin ( )˜ cu +cos ( )˜ cv )+ 1.(26)4.2.Kinetic control design4.2.1.Design uDifferentiating both sides of the first equation of (8)along the solutions of the first two equations of (6),and the fifth equation of (4)allows the control u to be designed as u =−m 11(k 4u e +k 5r 2u e −˙ u +b 2¯ e)−X u u −(m 22v +m 22s r)r −ˆX|u |u |u |u −cos ( )ˆ cu −sin ( )ˆ cv − M wu tanh (u e /( 5 3)),(27)1718K.D.Do,J.Pan/Automatica42(2006)1713–1722 where k4,k5are positive constants, 5is an ASPC to be specifiedlater.With(27),we have the u e dynamics as˙u e=−k4u e−k5r2u e−b2¯ e+m−111[X u u e+˜X|u|u|u|u+cos( )˜ cu+sin( )˜ cv+ wu(t)− M wu tanh(u e/( 5 3))].(28)4.2.2.Design rDifferentiating(22)along the solutions of(6),the fourth andsixth equations of(4)results in˙r e=j b1m33j rm32m22(m11ur−(Y r+Y|v|r|v|)r−(Y|r|v|r|+Y v+Y|v|v|v|)v+sin( ) cu−cos( ) cv− wv(t))−(m22v+m22s r)u+m11uv+(N v+N|v|v|v| +N|r|v|r|)v+(N r+N|v|r|v|+N|r|r|r|)r+ r+ cr+ wr(t)−j rj x e˙x e−j rj y e˙y e−j rj e˙e−j rj˙ −j rj s˙s−j rj u0˙u0−j rj˙u0¨u0−j rjˆ¯Y|v|v˙ˆ¯Y|v|v−j rjˆ cu˙ˆcu−j rjˆ cv˙ˆcv+j b1j x e˙x e+j b1j y e˙y e+j b1j e˙e+j b1j s˙s+j b1j u0˙u0+j b1jˆ¯Y|v|r˙ˆ¯Y|v|r+j b1jˆ¯Y|r|v˙ˆ¯Y|r|v−1m22j rj v−j b1j v[−m11ur+(Y r+Y|v|r|v|)r+(Y|r|v|r|+Y v+Y|v|v|v|)v−sin( ) cu+cos( ) cv+ wv(t)].(29)We now evaluate j b1/j r and make sure that this term is invert-ible.From the expression of b1(see(19)),we havej b1/j r=1−k2u0u−2d −1x e−m−122u−2d u0[m11 u+Y r+ˆ¯Y|v|r v(vr¯e/( 13))1+cosh2(¯ e vr/( 1 3))+ˆ¯Y|r|v v(vr¯e/( 23))1+cosh2(¯ e vr/( 2 3))+ˆ¯Y|v|r v tanh(¯ e vr/( 1 3))+ˆ¯Y|r|v v tanh(¯ e vr/ 2)].(30)A simple calculation shows thatj b1/j r 1−k2u−2d u0−u0m−122u2d(m11(k1+u0)+Y r+1.3(ˆ¯Y|v|r+ˆ¯Y|r|v)(|v+ 2|+ 2)).(31) Substituting(21)into(31)shows thatj b1/j r u∗20u−2d b∗1,if1st condition of Set I holds,j b1/j r b∗2,if1st condition of Set II holds,j b1/j r b∗3,if1st and2nd conditions of Set III holds.(32)Considering(32),we choose the control r from(29)asr=(m22v+m22s r)u−m11uv−(N v+ˆN|v|v|v|+ˆN|r|v|r|)v −(N r+ˆN|v|r|v|+ˆN|r|r|r|)r−ˆ cr− M wr tanh((j b1/j r)r e/( 6 3))−m32m−122(m11ur−(Y r+ˆY|v|r|v|)r−(ˆY|r|v|r|+Y v+ˆY|v|v|v|)v+sin( )ˆ cu−cos( )ˆ cv)+m33(j b1/j r)−1−k6r e+j rj x e˙x e+j rj y e˙y e+j rj e˙e+j rj˙ +j rj s˙s+j rj u0˙u0+j rj˙u0¨u0+j rjˆ¯Y|v|v˙ˆ¯Y|v|v+j rjˆ cu˙ˆcu+j rjˆ cv˙ˆcv−j b1j x e˙x e −j b1j y e˙y e−j b1j e˙e−j b1j s˙s−j b1j u0˙u0−j b1jˆ¯Y|v|r˙ˆ¯Y|v|r−j b1jˆ¯Y|r|v˙ˆ¯Y|r|v+1m22j rj v−j b1j v×(−m11ur+(Y r+ˆY|v|r|v|)r+(ˆY|r|v|r|+Y v+ˆY|v|v|v|)v−sin( )ˆ cu+cos( )ˆ cv)−m33 M wv 2tanh( 2r e/( 7 3),(33)where 2=j b1/j r+m32m−122m−133+m−122(j r/j v−j b1/j v); 6and 7are small positive constants to be specified later.Sub-stituting(33)into(29)results in˙r e=−k6r e−¯ e+(j b1/j r)m−133[m32m−122(−˜Y|v|r|v|r−˜Y|r|v|r|v −Y|v|v|v|v+sin( )˜ cu−cos( )˜ cv− wv(t))+˜N|v|v|v|v+˜N|r|v|r|v+˜N|v|r|v|r+˜N|r|r|r|r+˜ cr+ wr(t))−m−122×(j r/j v−j b1/j v)(˜Y|v|r|v|r+˜Y|r|v|r|+˜Y|v|v|v|v−sin( )˜ cu+cos( )˜ cv+ wv(t))− M wr tanh((j b1/j r)r e/( 6 3))− M wv 2tanh( 2r e/( 7 3))].(34) The estimatesˆX|u|u,ˆY|v|r,ˆY|r|v,ˆY|v|v,ˆN|v|v,ˆN|r|v,ˆN|v|r,ˆN|r|r,ˆcu,ˆ cv,ˆ cr are updated as follows:˙ˆX|u|u= 4proj(u e|u|u,ˆX|u|u),˙ˆY|v|r= 5proj(−r e m−122(m32m−133j b1/j r+(j r/j v−j b1/j v))|v|r,ˆY|v|r),˙ˆY|r|v= 6proj(−r e m−122(m32m−133j b1/j r+(j r/j v−j b1/j v))|r|v,ˆY|r|v),˙ˆY|v|v= 7proj(−r e m−122(m32m−133j b1/j r+(j r/j v−j b1/j v))|v|v,ˆ¯Y|v|v),˙ˆN|v|v= 8proj(r e m−133j b1/j r|v|v,ˆN|v|v),K.D.Do,J.Pan/Automatica42(2006)1713–17221719˙ˆN|r|v= 9proj(r e m−133j b1/j r|r|v,ˆN|r|v),˙ˆN|v|r= 10proj(r e m−133j b1/j r|v|r,ˆN|v|r),˙ˆN|r|r= 11proj(r e m−133j b1/j r|r|r,ˆN|r|r),˙ˆcu= 12proj(u e m −111cos( )+m−122(r e(m32m−133j b1/j r+j r/j v−j b1/j v)−u0u−2d¯ e)sin( ),ˆ cu),˙ˆcv= 13proj(u e m −111sin( )−m−122(r e(m32m−133j b1/j r+j r/j v−j b1/j v)−u0u−2d¯ e)cos( ),ˆ cu),˙ˆcr= 14proj(r e m −133j b1/j r,ˆ cr),(35)where i,4 i 14are positive constants to be specified later. It is noted that guaranteesˆX|u|u X|u|u(1+ |u|u),ˆY|v|r Y|v|r(1+ |v|r),ˆY|r|v Y|r|v(1+ |r|v),ˆY|v|v Y|v|v(1+ |v|v),ˆN|v|v N|v|v(1+ N|v|v),ˆN|r|v N|r|v(1+ N|r|v),ˆN|v|r N|v|r(1+ N|v|r),ˆN|r|r N|r|r(1+ N|r|r),ˆ cu M cu(1+ cu),ˆ cv M cv(1+ cv),ˆ cr M cr(1+ cr)(36) where |u|u, N|v|v, N|r|v, N|v|r, N|r|r, cu, cv, cr are ASPCs. We now present our main result whose proof is given in Ap-pendix A.Theorem1.Assume that Assumptions1and2hold,the con-trols u and r given by(27)and(33),and the adaptation laws given by(20),(24)and(35)solve the control objective. All signals of the closed loop system consisting of(15),(26), (28),(34)are bounded.Particularly,the transformed path-following errors(x e,y e,¯ e)globally asymptotically converge to a ball centered at the origin.The radius of this ball canbe adjustable by adjusting the control parameters k j,1 j 6, i,1 i 7, |v|r, |r|v,and the adaptation gains i,1 i 14. As a result,the actual position path following errors,(x−x d,y−y d),globally asymptotically converge to a ball cen-tered at the origin with an adjustable radius,and the ship will never turn around.Furthermore,the desired forward speed of the ship on the path can be adjusted by adjusting u0(t) and the total linear velocity of the ship is tangential to the path.5.Experimental resultsThis section describes the experimental setup and tested re-sults performed on a model ship(Fig.2)with a length of1.2m and a mass of17.5kg in the Swan river,Western Australia. The ship is equipped with:a differential global positioning sys-tem(DGPS)SF_2050G,one compass TCM2-50to obtain po-sition and orientation,two propellers,and two sets ofwirelessFig.2.An inside view of the model ship. transceivers to transmit and receive signals between the ship and the host computer.Based on our calculation and previ-ously executed empirical experiments,the parameters of our model ship are m11=25.8,m22=33.8,m33=2.76,m23= m32=6.2,D11=12+2.5|u|,D22=17+4.5|v|,D23=0.2, D32=0.5,D33=0.5+0.1|r|.These parameters are used in the controller and observer design,and implementation pre-sented in this section.All the nonlinear off-diagonal terms are ignored.A high-gain observer(Atassi&Khalil,1999)is used to estimate the velocities.We chose thehigh-gain observer be-cause the combination of global state feedback control with a high-gain observer allows for a separation approach where the high-gain isfirst designed to meet the control objectives,then the high-gain observer is designed,sufficiently fast to recover the performance achieved under state feedback.The high-gain observer is designed as follows:˙ˆ =J( )ˆv+H1( −ˆ ),M˙ˆv=−C(ˆv)ˆv−(D+D n(ˆv))ˆv+ +J T( )H2( −ˆ ),(37)whereˆ =[ˆxˆyˆ ]T andˆv=[ˆuˆvˆr]T are estimates of and v,respectively;H1=diag(ϑ−1,ϑ−1,ϑ−1)and H2=diag(h21ϑ−2,h22ϑ−2,h23ϑ−2)withϑa small positive constant,and h2i>1,i=1,2,3.The reference path is cho-sen as x d(s)=s,y d(s)=10sin(0.1s)+20,and u d0=0.6.The control design constants are k1=0.1,k2=0.05,k3=2,k4= 4,k5=0.1,k6=4, i=0.02,1 i 7, j=150,1 j 14, and all other“ ”are0.001.With this choice,Assumption1 holds with¯u∗d=10,u∗0=u M0=0.6,u∗01=u∗02=0,and Set II of Assumption2holds with b∗2=0.016, 2=4.5.The observer gains areϑ=0.1,h2i=3,i=1,2,3.The initial con-ditions are x(0)=−5m,y(0)=5m, (0)=0.009rad,u(0)= v(0)=0m/s,r(0)=0rad/s,ˆx(0)=−5m,ˆy(0)=5m,ˆ (0)=0.009rad,ˆu(0)=ˆv(0)=m/s,ˆr(0)=0rad/s.The es-timates of the nonlinear damping coefficients are initialized at 0.5of their values obtained by our previously executed empir-ical experiments as mentioned above.Experimental results are given in Figs.3and4where the ship position and orientation in the xy plane,and the position and orientation path following errors are plotted.These errors actually do not converge to zero mainly due to errors of the DGPS and compass.。