Robust adaptive control of nonholonomic mobile robot with parameter and nonparameter uncertainties

IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.2,APRIL2005261[24]P.Coelho and U.Nunes,“Lie algebra application to mobile robot con-trol:A tutorial,”Robotica,vol.21,no.5,pp.483–493,2003.[25]P.Coelho,“Path following control of wheeled mobile robots in presenceof uncertainties,”Ph.D.dissertation(in Portuguese),Univ.Coimbra, Coimbra,Portugal,2004.[26]G.Campion et al.,“Controllability and state feedback stabilization ofnonholonomic mechanical systems,”in Lecture Notes in Control and In-formation Science,C.C.de Wit et al.,Ed.New York:Springer-Verlag, 1991,vol.162,pp.106–124.[27]X.Yun and Y.Yamamoto,“Stability analysis of the internal dynamicsof a wheeled mobile robot,”J.Robot.Syst.,vol.14,no.10,pp.697–709, 1997.[28]L.E.Dubins,“On curves of minimal lenght with a constraint on av-erage curvature,and with prescribed initial and terminal positions and tangents,”Amer.J.Math.,vol.79,pp.497–516,1957.[29]H.Nijmeijer and A.Schaft,Nonlinear Dynamic Control Systems.NewYork:Springer-Verlag,1990.[30]K.J.Åström and B.Wittenmark,Computer Controlled Systems:Theoryand Design.Englewood Cliffs,NJ:Prentice-Hall,1997.[31]G.Pires and U.Nunes,“A wheelchair steered through voice commandsand assisted by a reactive fuzzy-logic controller,”J.Intell.Robot.Syst., vol.34,no.3,pp.301–314,2002.Robust Adaptive Control of Nonholonomic Mobile Robot With Parameter and Nonparameter UncertaintiesWenjie Dong and K.-D.KuhnertAbstract—This paper considers the tracking-control problem of a nonholonomic wheeled mobile robot with both parameter and nonpa-rameter uncertainties.A robust adaptive controller is proposed with the aid of the adaptive backstepping technique and the learning ability of neural networks.The proposed controller guarantees that the tracking error converges to a small ball containing the origin.The ball’s radius can be adjusted by control parameters.The proposed controller is successfully implemented in our simulator.Index Terms—Adaptive control,nonholonomic system,robust control, tracking control,wheeled mobile robot.I.I NTRODUCTIONIn recent years,there has been a growing interest in the design of feedback-control laws for mechanical systems subjected to non-holonomic constraints.Due to Brockett’s necessary condition for asymptotic feedback stabilization[1],a nonholonomic system cannot be asymptotically stabilized to a equilibrium by any smooth or con-tinuous pure-state feedback law.To overcome this limitation,several approaches have been proposed in last decade(see[18],[20],and the references therein).Tracking control is another control problem of nonholonomic sys-tems.Based on whether the system is described by a kinematic model or a dynamic model,the tracking-control problem is classified as ei-ther a kinematic or a dynamic tracking-control problem.The kine-matic tracking-control problem has been studied in recent years.With Manuscript received January1,2004;revised May11,2004.This paper was recommended for publication by Associate Editor G.Oriolo and Editor H.Arai upon evaluation of the reviewers’comments.W.Dong was with the Department of Electrical and Computer Engineering, University of Siegen,Siegen D-57068,Germany.K.-D.Kuhnert is with the Department of Electrical and Computer Engi-neering,University of Siegen,Siegen D-57068,Germany.Digital Object Identifier10.1109/TRO.2004.837236Fig.1.Configuration of a wheeled mobile robot.the aid of the linearization technique,local controllers are proposedin[9],[17],and[25].Based on the dynamic feedback linearizationand the differentialflatness concept,the dynamic controllers with sin-gular points are proposed in[4]and[10].In[15],global tracking con-trollers are proposed for nonholonomic wheeled mobile robots.Forchained systems,tracking controllers are proposed in[8]and[16].The dynamic tracking-control problem of the nonholonomic systemhas received attention recently,because most practical nonholonomicmechanical systems are dynamic systems and have uncertainties.In[23],Suand Stepanenko stu dy the tracking-control problem of the dy-namic nonholonomic systems with unknown inertia parameters,andpropose an adaptive controller.Chen et al.discuss the dynamic trackingproblem of the uncertain nonholonomic systems,and propose a ro-bust H-infinite controller in[2].In[9]and[11],the dynamic trackingproblem of a wheeled mobile robot is studied,and neural-network-based controllers are proposed.In[5],[6],[7],and[24],the dynamictracking-control problem of nonholonomic systems with uncertainty isdiscussed.Robust and adaptive controllers are proposed.Most of the results on the dynamic tracking problem of nonholo-nomic systems are proposed based on the assumption that the kine-matics of the system is exactly known,and there are only uncertaintiesin the dynamics of the system.However,in practice,there are uncer-tainties in both kinematics and dynamics.In this paper,we study thetracking-control problem of a wheeled mobile robot with uncertain-ties in both kinematics and dynamics.It is assumed that in the kine-matics,there are parameter uncertainties,and in the dynamics,there areparameter uncertainties and nonparameter uncertainties.To solve thistracking-control problem,a robust adaptive neural-network controlleris proposed with the aid of backstepping techniques and the approxi-mation property of neural networks.II.P ROBLEM S TATEMENTConsider a wheeled mobile robot moving on a horizontal plane(Fig.1).The robot is composed of a rigid body,twofixed rear wheels,and one steering wheel.Two torque motors are equipped in the frontwheel for driving and steering.Given a differentiable simple curve(C)defined by one of its points,the unitary tangent vector at this point,andits curvature curv(s),with s being the curvilinear coordinate alongthe curve,for a point Q in the curve(C),assume that the curvilinearcoordinate at Q is s.Let f Q;T(s);N(s)g be the Frenet frame onthe curve at point Q,and assume that j curv(s)j<1=R(8s)whereR(>0)is a constant.If the distance between P and the curve(C)issmaller than R,the position of P is parameterized by(s;d),whered is the coordinate of P along N(s).The robot’s configuration isparameterized by q=[q1;q2;q3;q4]T=[s;d; ; ]T,where is theangle between P F and T(s)and is the steering angle of the front 1552-3098/$20.00©2005IEEEAuthorized licensed use limited to: SOUTH CHINA UNIVERSITY OF TECHNOLOGY. 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Restrictions apply.262IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.2,APRIL2005 wheel with respect to the robot body.By the classic law of mechanics,and also the results in[22],onehas_q1=v1cos q310curv(q1)q2 _q2=v1sin q3_q3=v1tan q4l 0v1curv(q1)cos q3 10curv(q1)q2_q4=v2(1)M(q)_v+C(q;_q)v+G(q)+U(q;_q)=B(q) (2) where l is the distance between the two points P and F;v1is the ve-locity of the point P;v2is the angular velocity of the steering wheel, M(q)is a bounded positive definite symmetric inertia matrix,C(q;_q)_q is centripetal and Coriolis torques,G(q)is the gravitational torque, U(q;_q)includes the unmodeled dynamics of the system and bounded continuous external disturbance,B(q)is the input matrix, is the con-trol input,and the superscript T denotes the transpose.For(1)–(2),it is assumed that:1)in(1),l is not exactly known,i.e.,l2[l min;l max]wherel max(>0)and l min(>0)are known;2)in(2),the matrices M(q);C(q;_q);G(q),and U(q;_q)areunknown;3)in(2),the expression of B(q)is known.In fact,it can beeasily derived that B(q)=diag[1=(r cos q4);1].In B(q),itis assumed that r is not exactly known,i.e.,r2[r min;r max]where r min(>0)and r max(>0)are known.Control Problem:Given a desired simple curve(C)and a desired velocity v31(t)of the robot,for(1)–(2),the control problem in this paper is defined asfinding a controller ,such thatlim t!1q2(t)=0limt!1q3(t)=0limt!1(v1(t)0v31(t))=0:Remark1:Noting that the assumption j curv(q1)j<1=R,(1)is well-defined if j q2j<R and j q4j< =2.In the controller design, these conditions will be guaranteed if j q2(0)j<R and j q4(0)j< =2. In(2),the following well-known property is satisfied[23]. Property1:For a suitably defined C(q;_q);(_M02C)is skew-symmetric.III.B ACKSTEPPING D ESIGN P ROCEDURETo deal with parameter uncertainties and nonparameter uncertainties in(1)–(2),the adaptive backstepping technique[19]and the learning ability of neural networks[3],[14]are used to design the controller. Assuming j q2(0)j<R;j q3(0)j< =2,and j q4(0)j< =2, let u1=v1cos q3=(10curv(q1)q2);u2=v2;b=1=l;g2= [0;0;0;1]T;g1=[1;(10curv(q1)q2)tan q3;0curv(q1);0]T;g3= [0;0;(10curv(q1)q2)tan q4=cos q3;0]T,(1)–(2)can be written as_q=g1u1+g2u2+bg3u1(3)M1(q)_u+C1(q;_q)u+G1(q)+U1(q;_q)=B1(q) (4) whereM1(q)=9T(q)M(q)9(q)C1(q;_q)=9T(q)(M(q)_9(q)+C(p;_p)9(q))G1(q)=9T(q)G(q)U1(q;_q)=9T(q)U(q;_q)B1(q)=diag[(10curv(q1)q2)=(r cos q3cos q4);1]9(q)=diag[(10curv(q1)q2)=cos q3;1]:Step1:Introducing~u1=u10v31if u1were the actual control input,one had~u1 0and u1 v31.Letz2=h(q2)where h(q2)is a smooth monotonic function which maps(0R;R) onto(01;+1),with thefirst derivative(with respect to q2)strictly larger than a positive real number,and such that h(0)=0,then _z2=v31L g z2+u2L g z2+bv31L g z2=v31L g z2(5) where L g z j is the Lie derivative of z j along g i.Hereafter,L means Lie derivative in this paper.Introducingz3=L g z20 3if L g z2were the actual control input,one had z3 0and L g z2 3.Let Lyapunov functionV2=12z22to make_V2=0k2z22v31we choose3=0k2z2(6) where constant k2(>0)is a design parameter.Since L g z2is not the control,z3 0.So_V2=0k2z22v31+z2z3v31:The second term z2z3v31will be cancelled at the next step.The closed-loop system(5)with(6)is_z2=0k2z2v31+z3v31(7)_z3=v31L2g z20k22z2v31+k2z3v31+bv31L g L g z2:(8) Step2:Introducingz4=L2g z20 4let Lyapunov functionV3=12z22+z23+12011(^b0b)2where constant 1(>0)is a design parameter,then_V3=0k2z22v31+z3(z4+z20k22z2+k2z3+^bLg L g z2+ 4)v31+ 011(^b0b)(_^b0 1v31z3L g L g z2): If L2g z2were the actual control input,one had z4 0.To make_V3=0k2z22v310k3z23v31we would choose_^b=14=0(k2+k3)z30(10k22)z20^bL g L g z2(9) where1= 1v31z3L g L g z2constant k3(>0)is a design parameter.Authorized licensed use limited to: SOUTH CHINA UNIVERSITY OF TECHNOLOGY. 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Restrictions apply.IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.2,APRIL 2005263Since L 2g z 2is not the control,z 4 0and we do not use _^b = 1asan update law in the control.Then_V 3=0k 2z 22v 310k 3z 23v 31+z 3z 4v 31+ 011(^b 0b )(_^b 0 1):The third term z 3z 4v 31will be cancelled at the next step.The closed-loop system (8)with (9)is_z 3=0k 3z 3v 31+z 4v 310z 2v 31+(b 0^b )v 31L g L g z 2_z 4=v 31L 3g z 2+(k 2+k 3)z 4v 310(k 2k 3+k 2301+k 22)z 3v 310(2k 20k 32+k 3)z 2v 31+_^bL g L g z 2+^bv 31L g L g L g z 2+(k 2+k 3)(b 0^b )v 31L g L g z 2+^bbv 31L 2g L g z 2+bv 31L g L 2gz 2+^bu 2L g L g L g z 2:(10)Step 3:Introducing~u 2=u 20 5let Lyapunov functionV 4=12z 22+z 23+z 24+12011(^b 0b )2then _V4=0k 2z 22v 310k 3z 23v 31+z 4[z 3v 31+v 31L 3g z 2+(k 2+k 3)z 4v 310(k 2k 3+k 2301+k 22)z 3v 310(2k 20k 32+k 3)z 2v 31+_^bL g L g z 2+^bv 31L g L g L g z 2+^b 2v 31L 2g L g z 2+^bv 31L g L 2gz 2+^b (~u 2+ 5)L g L g L g z 2]+ 011(^b 0b )(_^b 0 1(z 3v 31L g L g z 2+z 4[(k 2+k 3)2v 31L g L g z 2+^bv 31L 2g L g z 2+v 31L g L 2g z 2])):If u 2were the actual control input,one had ~u 2 0.The update law of^b is chosen as_^b= 20 1(^b 0b 0);if ^b 2(b l ;b u );or ^b =b l ; 2 0;or ^b =b u 2 00 1(^b 0b 0);if ^b =b l ; 2<0or ^b =b u 2>0(11)where b l =1=l max and b u =1=l min ,constants 1(>0)and b 0(2(b l ;b u ))are design parameters,and 2is defined by 2= 1+1z 4v 31[(k 2+k 3)L g L g z 2+^bL 2g L g z 2+L g L 2g z 2]:The virtual control 5is chosen as5=[0k 4z 4v 310z 3v 310v 31L 3g z 20(k 2+k 3)z 4v 31+(k 2k 3+k 2301+k 22)z 3v 31+(2k 20k 32+k 3)z 2v 31+50^bv 31L g L g L g z 20^b 2v 31L 2g L g z 20^bv 31L g L 2gz 2]=(^bL g L g L g z 2)where 5=1+ 22+ 11+(^b 0b 0)2L g L g z 22tanh z41+ 22+ 11+(^b 0b 0)2L g L g z 2)= 2)constants k 4(>0)and 2(>0)are design parameters,and 5is thethird stabilizing function.Then_V4 0k 2z 22v 310k 3z 23v 310k 4z 24v 310 011 12(^b 0b )2011 12(^b 0b 0)2+ 011 12(b 0b 0)2+ 2where satisfies =e0( +1)(i.e., =0:2785)[21].Remark 2:By the relation between z and q ,boundedness of z 2;z 3,and z 4guarantees that q 22(0R;R );q 32(0 =2; =2),and q 42(0 =2; =2).The update law (11)guarantees ^b 2[b l ;b u ]all the time.Therefore, 5is well defined.Since u 2is not the control input,~u 2 0.However,we will choose(11)as the final update law of ^b in the control.Then_V 4 0k 2z 22v 310k 3z 23v 310k 4z 24v 31+z 4^bL g L g L g z 2~u20 011 12(^b 0b )20 011 12(^b 0b 0)2+ 011 12(b 0b 0)2+ 2:The closed-loop system (10)is_z 4=0k 4z 4v 310z 3v 31+~u 2^bL g L g L g z 2+L g L g z 2_^b +51+(b 0^b )[^bL 2gL g z 2+L g L 2g z 2+(k 2+k 3)L g L g z 2]v 31:Step 4:Since u 1is not the control input,~u 1 0.Let ~u =[~u 1;~u 2]Tand =[v 31; 5]T(12)thenM 1_~u =B 1 0C 1~u 0(M 1_ +C 1 +G 1+U 1):(13)In (13),M 1;C 1;G 1,and U 1are unknown.Also in B 1;r is un-known.To deal with the uncertainty of (M 1_ +C 1 +G 1+U 1),a two-layer neural network with sigmoid base functions will be usedto approximate it.Since (M 1_+C 1 +G 1+U 1)is continuous,let s (q;_q;;_ )be an m -vector of continuous sigmoidal functions,ac-cording to the approximation property of neural networks [3]M 1(q )_ +C 1(q;_q ) +G 1(q )+U 1(q;_q )=w T s (q;_q;;_ )+ (14)where w 2R m 22is an unknown optimal constant weight vector,andis defined by w :=argmin 2Rfsup (q;_q; ;_ )2k M 1(q )_ +C 1(q;_q )+G 1(q )+U 1(q;_q )0 T s (q;_q;;_ )kg where is a compact region. =[ 1; 2]T represents the networkreconstruction error corresponding to the optimal weight vector.Gen-erally,increasing the number of adjustable weights (i.e.,m )reduces the network reconstruction error.The approximation results for neural networks indicate that if m is sufficiently large,then can be made bounded on a compact region [3],[12].Given a compact region ,we assume thatk k ;8(q;_q; ;_ )2(15)where is an unknown bound.It is worth noting that the unknown bound is not unique,since any constant which is greater than satisfies (15).To avoid confusion,we define to be the smallest constant,such that (15)is satisfied.With (14),(13)can be written asM 1(q )_~u =B 1(q ) 0C 1(q;_q )~u 0w T s 0 :Defining c =1=r ,and letting ^c ;^w ,and ^be the estimates of c;w ,and ,respectively,we choose the control law=^B 011[0K p ~u +^w T s (q;u )0^tanh ~u2+3](16)Authorized licensed use limited to: SOUTH CHINA UNIVERSITY OF TECHNOLOGY. 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Restrictions apply.264IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.2,APRIL2005where K p is a positive definite matrix,and^B1is the value ofB1corresponding to the estimate^c(i.e.,^B1=diag[^c(10curv(q1)q2)=cos q3cos q4;1]).3and the update laws of^w and^will be determined next.ThenM1_~u=0K p~u0C1~u+(^w0w)T s0^tanh~u20 +3+(B10^B1) :LetV5=12z22+z23+z24+~u T M1~u+ 011(^b0b)2+ 012tr2((^w0w)T(^w0w))+ 013(^0)2+ 014(^c0c)2 then_V5=0k2z22v310k3z23v310k4z24v31+z4L g L g z2_^b+z45+ 011(^b0b)_b0z3u1L g L g z20z4u 1^bL2g L g z2 +L g L2g z2+(k2+k3)L g L g z20~u T K p~u+tr(^w0w)T 012_^w+s~uT+0~u1tanh~u120~u1 10~u2tanh~u220~u22+ 013(^0)2_^0 3~u Ttanh~u2+~u T30k2z22+k3z23+k4z24~u1+z 4^bL g L g L g z2+L g L2g z2~u2+ 014(^c0c )_^c0 4~u1 1(10curv(q1)q2)cos q3cos q4where 1is thefirst element of .If we choose the update law^b as in(11),the update laws of^w;^, and^c are as follows:_^w=02s~u T0 1(^w0w0)(17)_^=3~u Ttanh~u20 1(^00)(18)_^c=4~u1 1(10curv(q1)q2)cos q3cos q40 1(^c0c0);if^c2(c l;c u)or^c=c l~u1 1(10curv(q1)q2)cos q3cos q4>0or^c=c u~u1 1(10curv(q1)q2)cos q3cos q4<00 1(^c0c0);if^c=c l;~u1 1(10curv(q1)q2)cos q3cos q40 or^c=c u;~u1 1(10curv(q1)q2)cos q3cos q4(19)3=k2z22+k3z23+k4z24 0z4^bL g L g L g z2where 2(>0); 3(>0); 4(>0);w0;0,and c0(2(c l;c u))are design parameters,c l=1=r max,and c u=1=r min,then_V5 0k2z22v310k3z23v310k4z24v310 011 12(^b0b)20~u T K p~u0 01221tr((^w0w)T(^w0w))0 0132 1(^0)20 01421(^c0c)2+01112(b0b0)2+ 01221tr((w0w0)T(w0w0))+01321(00)2+ 01421(c0c0)2+2 2+ 2:(20)IV.M AIN R ESULTS AND D ISCUSSIONSA.Main ResultsWith the aid of the preceding design procedure,one has the followingresult.Theorem1:With the controller(16)and the update laws of^b;^w;^,and^c defined in(11),(17),(18),and(19),respectively,if v31(t)v>0,then z i(2 i 4);~u;(^b0b);(^c0c);(^w0w),and(^0)are uniformly bounded,and exponentially converge to a smallball containing the origin.The radius of the ball can be adjusted by thedesign parameters.Proof:It can be proved that the modified projection algorithm(11)guarantees that^b2[b l;b u],therefore 5is well defined all thetime.With the update law(19),it can be proved that^c2[c l;c u].So the control law(16)is well defined.Therefore,all variables in thesystem are well defined.Differentiating V5with respect to time alongthe closed-loop system,one has(20).Therefore_V5 0 1V5+ 2(21)where 1is a positive constant which depends on the control parame-ters,and2=01112(b0b0)2+01221tr((w0w0)T(w0w0))+01321(00)2+01421(c0c0)2+2 2+ 2:SoV5(t) V5(0)0 21e0 t+ 21:z i(2 i 4);~u;^b;^c;^w,and^are uniformly bounded and expo-nentially converge to a small ball.The radius of the ball can be adjustedby the design parameters i(1 i 4); 1;b0;c0;w0;0,and 2.With the aid of the state transformation and Theorem1,one has thefollowing result.Theorem2:With the controller(16)and the update laws^b;^w;^,and^c defined in(11),(17),(18),and(19),respectively,if j q2(0)j<R;j q3(0)j= =2;j q4(0)j= =2,and v31(t) v>0,then q i(2i 4);(v10v31);(^b0b);(^c0c);(^w0w),and(^0)are uniformlybounded and converge to a small ball containing the origin.The radiusof the ball can be adjusted by the design parameters.Proof:By Theorem1,z i(2 i 4);~u;(^b0b);(^c0c);(^w0w),and(^0)are uniformly bounded and exponentially converge toa small ball.By calculation,it can be proved that q i(2 i 4)and(v10v31)are uniformly bounded and converge to a small ball.B.DiscussionsIf j q2(0)j<R,in order to make q22(0R;R)all the time,z2=h(q2)is introduced in Step1.With the condition imposed on h(q2),ifz2is bounded,q22(0R;R).Therefore,the definition of point Q isunique,and d is well defined in the control.If R<1,one choice ofh(q2)ish(q2)=2Rtan q22R:Specially,if curv(s)=0,one can choose h(q2)=q2.If j q3(0)j<=2and j q4(0)j< =2,the proposed controller will make j q3j<=2and j q4j< =2all the time.If q2(0)j R,j q3(0)j= =2,orj q4(0)j= =2,one canfirst use an open-loop control law to makethe robot move into the region where the proposed controller can beapplied,then apply the proposed controller.Unknown parameters b(=1=l)and c(=1=r)are updated by theadaptive laws(11)and(19),respectively.They guarantee that^b2[b l;b u]and^c2[c l;c u].The neural network is used to approximateAuthorized licensed use limited to: SOUTH CHINA UNIVERSITY OF TECHNOLOGY. 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Restrictions apply.IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.2,APRIL 2005265Fig.2.Response of d.Fig.3.Response of .the unknown dynamics of the system.The weight matrix is updated bythe adaptive law (17).The reconstruction error of the neural network is estimated by the adaptive law (18).The control parameters are k i (2 i 4);K p ; i (1 i 4); 1; 2;b 0;c 0; 0,and w rge values of k i (2 i 4)and K p make q i (2 i 4)and (v 10v 31)converge quickly to the small ball.Small values of 01i 1(1 i 4)and 1make the radius of the ball small.Parameters b 0;c 0; 0,and w 0also affect the radius of the small ball.If b 0;c 0; 0,and w 0are close to b;c; ,and w ,respectively,the tracking error will be small.Therefore,in order tomake q i (2 i 4)and (v 10v 31)converge quickly to the origin,one can make k i (1 i 4);K p ; i (1 i 4)large and 1small.V .S IMULATIONIn order to verify the validity of the proposed controller,we imple-ment the controller in the simulator available in our lab.The setup of the simulator is comprised of two interconnected computers.One com-puter,the so-called System-PC,processes the video streams and calcu-lates the control commands for the robot to follow the path.These con-trol commands are then passed to the second computer,the so-called Simulator-PC,which is responsible for the robot and environment sim-ulation.In the experiment,we choose one type of wheeledmobileFig.4.Response of (v 0v ).Fig.5.Response of ^b.Fig.6.Response of ^c .robots and a rocky path to test the controller.Figs.2–4show the re-sponses of d; ,and (v 10v 31)with a group of control parameters.From the results,it is shown that d; ,and (v 10v 31)converge to zero during the control.Figs.5and 6show the responses of ^b and ^c .It is Authorized licensed use limited to: SOUTH CHINA UNIVERSITY OF TECHNOLOGY. 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Restrictions apply.266IEEE TRANSACTIONS ON ROBOTICS,VOL.21,NO.2,APRIL 2005shown that they are bounded.Especially,^b and ^c do not go throughzero.Responses of ^ and ^w also show the boundedness of ^ and ^w .The control inputs calculated from the control law are bounded and not large.They can be realized by typical mobile actuators.These exper-imental results show that the proposed controller is effective.To test the proposed controller further,we implement it in different robots and different paths.All results show that the controller is effective.VI.C ONCLUSIONIn this paper,the tracking-control problem of a nonholonomic wheeled mobile robot with parameter uncertainties and nonparameter uncertainties is considered.A robust adaptive controller is proposed.The idea in this paper can be applied to control other uncertain nonholonomic dynamics systems.R EFERENCES[1]R.W.Brockett,“Asymptotic stability and feedback stabilization,”in Dif-ferential Geometric Control Theory ,R.W.Brockett,Ed.Boston,MA:Birkhauser,1983,pp.181–208.[2] B.S.Chen et al.,“A robustH model reference tracking design fornonholonomic mechanical control systems,”Int.J.Control ,vol.63,pp.283–306,1996.[3]G.Cybenko,“Approximation by superpositions of a sigmoidal func-tion,”Math.Control Signals Syst.,vol.2,no.4,pp.303–314,1989.[4] B.d’Andrea-Novel et al.,“Control of nonholonomic wheeled mobilerobots by state feedback linearization,”Int.J.Robot.Res.,vol.14,pp.543–559,1995.[5]W.Dong and W.L.Xu,“Adaptive tracking control of uncertain non-holonomic dynamic system,”IEEE Trans.Autom.Control ,vol.43,pp.450–454,Mar.2001.[6]W.Dong et al.,“Tracking control of dynamic nonholonomic systemsand its application to mobile robots with uncertainty,”IEEE Trans.Robot.Autom.,vol.16,pp.870–874,Dec.2000.[7]W.Dong et al.,“Trajectory tracking control of dynamic nonholonomicsystems with unknown dynamics,”Int.J.Robust,Nonlinear Control ,vol.9,no.13,pp.905–922,1999.[8]W.Dong and W.Huo,“Trajectory tracking control of chained systems,”Zidonghua Xuebao/Acta Automatica Sinica ,vol.26,no.3,pp.310–316,2000.[9]R.Fierro and F.L.Lewis,“Control of a nonholonomic mobile robotusing neural networks,”IEEE Trans.Neural Netw.,vol.9,pp.589–600,Jul.1998.[10]M.Fliess et al.,“Design of trajectory stabilizing feedback for driftlessflat systems,”in Proc.Eur.Control Conf.,1995,pp.1882–1887.[11]T.Fukao,H.Nakagawa,and N.Adachi,“Adaptive tracking control ofa nonholonomic mobile robot,”IEEE Trans.Robot.Autom.,vol.16,pp.609–615,Oct.2000.[12]S.S.Ge et al.,Stable Adaptive Neural Network Control .Boston,MA:Kluwer,2001.[13]S.S.Ge et al.,“Adaptive robust stabilization of dynamic nonholonomicchained systems,”J.Robot.Syst.,vol.18,no.3,pp.119–113,2001.[14]K.Hornik et al.,“Multilayer feedforward networks are universal approx-imators,”Neural Netw.,vol.2,pp.359–366,1989.[15]Z.-P.Jiang and H.Nijmeijer,“Tracking control of mobile robots:A casestudy in backstepping,”Automatica ,vol.33,no.7,pp.1393–1399,1997.[16],“A recursive technique for tracking control of nonholonomicsystems in chained form,”IEEE Trans.Autom.Control ,vol.44,pp.256–279,Feb.1999.[17]Y .Kanayama et al.,“A stable tracking control method for an au-tonomous mobile robot,”in Proc.IEEE Conf.Robot.Autom.,1990,pp.384–389.[18]I.Kolmanovsky and N.H.McClamroch,“Development in nonholo-nomic control problems,”IEEE Control Syst.Mag.,vol.15,pp.20–36,Dec.1995.[19]M.Krstic et al.,Nonlinear and Adaptive Control Design .New York:Wiley,1995.[20]P.Lucibello and G.Oriolo,“Robust stabilization by iterative statesteering with an application to chained-form systems,”Automatica ,vol.37,no.1,pp.71–79,2001.[21]M.M.Polycarpou,“Stable adaptive neural network control scheme fornonlinear systems,”IEEE Trans.Autom.Control ,vol.41,pp.447–451,Mar.1996.[22] C.Samson,“Path following and time-varying feedback stabilization ofwheeled mobile robot,”in Proc.ICARCV ,vol.1,Singapore,1992,pp.RO-13.1.1–RO-13.1.5.[23] C.Y .Suand Y .Stepanenko,“Robu st motion/force control of mechanicalsystems with classical nonholonomic constraints,”IEEE Trans.Autom.Control ,vol.39,pp.609–614,Mar.1994.[24]M.Oya,C.Y .Su,and R.Katoh,“Robust adaptive motion/force trackingcontrol of uncertain nonholonomic mechanical systems,”IEEE Trans.Robot.Autom.,vol.19,pp.175–181,Feb.2003.[25]G.Walsh et al.,“Stabilization of trajectories for systems with nonholo-nomic constraints,”IEEE Trans.Autom.Control ,vol.39,pp.216–222,Jan.1994.A New Formulation of Visual Servoing Based on Cylindrical Coordinate SystemMasami Iwatsuki and Norimitsu OkiyamaAbstract—Image-based visual servoing is a flexible and robust technique to control a robot and guide it to a desired position only by using two-di-mensional visual data.However,it is well known that the classical visual servoing based on the Cartesian coordinate system has one crucial problem,that the camera moves backward at infinity,in case that the camera motion from the initial to desired poses is a pure rotation of 180around the op-tical axis.This paper proposes a new f ormulation of visual servoing,based on a cylindrical coordinate system that can shift the position of the origin.The proposed approach can interpret from a pure rotation around an ar-bitrary axis to the proper camera rotational motion.It is shown that this formulation contains the classical approach based on the Cartesian coor-dinate system as an extreme case with the origin located at infinity.Fur-thermore,we propose a decision method of the origin-shift parameters by estimating a rotational motion from the differences between initial and de-sired image-plane positions of feature points.Index Terms—Camera retreat problem,cylindrical coordinate system,eye-in-hand robot,visual servoing.I.I NTRODUCTIONVisual servoing is a flexible and robust control technique of robots using vision in feedback-control loops.It is classified into position-based approaches and image-based approaches [1],[2].It is known that the image-based visual servoing can derive a robot motion directly from two-dimensional (2-D)visual data [3].One of the chief advantages to the image-based visual servoing is that the positioning accuracy of the robot is less sensitive to robot and camera calibration errors and image measurement errors.In most of the image-based visual servo systems,image features are composed with a set of image points,frequently the centroids of re-gions,thanks to the robustness and implementation ease of feature ex-traction [1].Unfortunately,the conventional approach of monocularManuscript received April 5,2003;revised May 11,2004;June 19,2004;July 19,2004.This paper was recommended for publication by Associate Editor F.Chaumette and Editor S.Hutchinson upon evaluation of the reviewers’com-ments.This paper was presented in part at the IEEE/RSJ International Confer-ence on Intelligent Robots and Systems,Lausanne,Switzerland,October 2002.The authors are with the Faculty of Engineering,Hosei University,Tokyo 184-8584,Japan (e-mail:iwatsuki@k.hosei.ac.jp).Digital Object Identifier 10.1109/TRO.2004.8372421552-3098/$20.00©2005IEEEAuthorized licensed use limited to: SOUTH CHINA UNIVERSITY OF TECHNOLOGY. 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