Tracking and Regulation Control of a Mobile Robot System With Kinematic Disturbances: A Variable Structure-Like Approach

Transcription

1 W. E. Dixon D. M. Dawson E. Zergeroglu Department of Electrical & Computer Engineering, Clemson University, Clemson, SC Tracking an Regulation Control of a Mobile Robot System With Kinematic Disturbances: A Variable Structure-Like Approach This paper presents the esign of a variable structure-like tracking controller for a mobile robot system. The controller provies robustness with regar to boune isturbances in the kinematic moel. Through the use of a ynamic oscillator an a Lyapunov-base stability analysis, we emonstrate that the position an orientation tracking errors exponentially converge to a neighborhoo about zero that can be mae arbitrarily small (i.e., the controller ensures that the tracking error is globally uniformly ultimately boune (GUUB)). In aition, we illustrate how the propose tracking controller can also be utilize to achieve GUUB regulation to an arbitrary esire setpoint. An extension is also provie that illustrates how a smooth, time-varying control law can be utilize to achieve setpoint regulation espite parametric uncertainty in the kinematic moel. Simulation results are presente to emonstrate the performance of the propose controllers. S Introuction Over the past twenty years the control of wheele mobile robots WMRs has been heavily stuie see,, an the references therein for an in-epth review of the previous workue to the challenging theoretical nature of the problem i.e., WMRs are nonlinear, uneractuate systems subject to nonholonomic constraints impose by a pure rolling an nonslipping assumption an the wie range of applications that are well suite to their use e.g., munitions hanling, exploration, security, an monitoring, etc.. Several researchers have examine the regulation 3,4 an the tracking control 5 problems for WMRs that are subject to isturbances in the kinematic moel that violate the pure rolling an nonslipping assumption. Specifically, a quasi-sliing moe controller was presente by Canuas e Wit et al. in 3 that achieve exponential regulation of the position/orientation of a WMR subject to either a constant matche isturbance or a state vanishing isturbance that violates the nonholonomic constraint. In 4, Corraini et al. propose a iscrete-time, quasi-sliing moe controller that regulate the position of a WMR subject to a similar isturbance to a neighborhoo about the origin. Note that the quasi-sliing moe regulation controllers presente in 3,4 are not ifferentiable, an unfortunately, the stanar backstepping proceure, often use for incorporating the mechanical ynamics, requires that the kinematic controller be ifferentiable see the iscussion in. In 6, Anrea-Novel et al. propose a singular perturbation formulation that le to robustness results for feeback linearizing control laws with sufficiently small slipping an skiing effects. In 5, Leroquais et al. use the results in 6 to esign a linear, ifferentiable time-varying feeback law that achieve local uniformly asymptotically stable tracking of a time-varying reference trajectory; however, ue to restrictions on the reference trajectory, the tracking controller cannot be applie to the regulation problem. It shoul be note that the controller propose in 5 inclue the ynamic moel of the WMR. In aition to problems concerning isturbances in the kinematic moel, researchers have also investigate the effects of parametric Contribute by the Dynamic Systems an Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receive by the Dynamic Systems an Control Division February 4, 000. Associate Technical Eitors: E. Misawa an V. Utkin. uncertainty in the kinematic moel. For example, 7 9 examine regulating a WMR with uncertain parameters multiplie by the control inputs of the kinematic moel. Specifically, in 7, Hespanha et al. utilize a supervisory control strategy to switch between a suitably efine family of caniate control laws to solve the so-calle parking problem for a WMR. In 8, Jiang propose a switching controller to exponentially regulate a WMR, an in 9, Jiang extene the results to the general chaine form. In this paper, we present the esign of a variable structure-like tracking controller for a mobile robot system that is subject to kinematic isturbances. The kinematic isturbances consiere in this paper represent a broaer class of isturbances than previously examine i.e., the kinematic moel presente in 3,4 are special cases of the kinematic moel examine in this paper. Through the use of a ynamic oscillator an a Lyapunov-base stability analysis, we show that the position an orientation tracking errors exponentially converge to a neighborhoo about zero that can be mae arbitrarily small i.e., the controller ensures that the tracking error is globally uniformly ultimately boune GUUB. In aition, since we only require that the reference trajectory be boune, the propose tracking controller can also be utilize to achieve GUUB regulation; hence, a unifie control framework for both the tracking an the regulation problem is propose. Finally, we also illustrate how the ifferentiable, timevarying controller presente in can be utilize to solve the regulation problem of a WMR with the same uncertain kinematic moel as examine in 7 9. The paper is organize as follows. In Section, we evelop the kinematic moel of a WMR that is subject to kinematic isturbances an then transform the moel into a form which facilitates the subsequent control evelopment. In Section 3, we present the control law an the corresponing close-loop error system. In Section 4, we provie a Lyapunov base stability analysis that illustrates global uniformly ultimately boune tracking. In Section 5, we present a setpoint extension for the propose controller. In Section 6, we present an extension regaring the setpoint regulating problem for a WMR with uncertain parameters in the kinematic moel. In Section 7, we provie simulation results to illustrate the performance of the propose controllers, an in Section 8, we present some concluing remarks. 66 Õ Vol., DECEMBER 000 Copyright 000 by ASME Transactions of the ASME

2 Problem Formulation. WMR Kinematic Moel. The kinematic moel for the so-calle kinematic wheel is given as follows q S q v t t 3 t T () where q(t), q (t) R 3 are efine as q x c y c T q ẋ c ẏ c T () x c (t) an y c (t) enote the Cartesian position of the center of mass COM of the WMR along the X an Y-coorinate axis of the Cartesian plane see Fig., (t) R represents the orientation of the WMR see Fig., ẋ c (t), ẏ c (t) enote the Cartesian components of the linear velocity of the COM, (t) R enotes the angular velocity of the COM, the matrix S(q) R 3 is efine as follows S q cos 0 sin 0 0, (3) the velocity vector v(t) R is efine as follows v v v T v l T (4) with v l (t) R enoting the linear velocity of the COM, an (t), (t), 3 (t) R represent unknown isturbances that are assume to be upper boune as shown below t, t, 3 t 3 (5) where,, 3 R are positive bouning constants. Note that if (t), (t), 3 (t) 0, the stanar kinematic moel for the pure rolling an nonslipping kinematic wheel is recovere. Remark. Note that the kinematic moel for a WMR subject to the so-calle matche isturbance is given as follows [3] q S q v M t cos sin, 0 T (6) where M (t) R enotes a boune isturbance. In aition, the kinematic moel for a WMR subject to the so-calle unmatche isturbance is given as follows [3] q S q v U t sin cos 0 T (7) where U (t) R enotes a boune isturbance. Note that in orer to obtain the exponential regulation result presente in [3], the unmatche isturbance U (t) must be upper boune by a function of the states, whereas the GUUB result obtaine in [4] require the isturbance be upper boune by a constant. From a control point of view, it is easy to see from (), (6), an (7) that the matche isturbance an unmatche isturbance problems are both special cases of the moel use in ().. Moel Transformation. As efine in previous work e.g., see 0 an, the time-varying reference trajectory for the WMR is generate via a reference robot which moves accoring to the following ynamic trajectory q r S q r v r (8) where S( ) was efine in 3, q r (t) x rc (t) y rc (t) r (t) T R 3 enotes the esire time-varying position/orientation trajectory, an v r (t) v r (t) v r (t) T R enotes the reference timevarying linear/angular velocity. With regar to 8, it is assume that the signal v r (t) is constructe to prouce the esire motion an that v r (t), v r(t), q r (t), an q r(t) are boune for all time. To facilitate the subsequent control synthesis an the corresponing stability proof, we efine the following global invertible transformation w cos sin sin cos 0 z 0 0 z ỹ cos sin 0 x where w(t) R an z(t) z (t) z (t) T R are auxiliary tracking error variables an x (t),ỹ(t), (t) R enote the ifference between the actual Cartesian position an orientation of the COM an the reference position an orientation of the COM as follows (9) x x c x rc ỹ y c y rc r. (0).3 Open-Loop Tracking Error Development. After taking the time erivative of 9, an using 4, 8 0 we can rewrite the open-loop tracking error ynamics in a more convenient form as follows ẇ u T J T z f ż u () where J R is a skew-symmetric matrix efine as J 0 () 0 f (z,v r,t) R is an auxiliary signal efine as f v r z r sin z (3) the auxiliary kinematic control input, enote by u(t) u (t) u (t) T R, is efine in terms of the WMR position/ orientation, the WMR linear velocities, an the reference trajectory as follows u T v v r v r cos Fig. Two-imensional kinematic wheel v Tu v r cos v r x sin ỹ cos v r (4) the auxiliary matrix T R is efine as follows x sin ỹ cos T (5) 0 an (t) R an (t) R are auxiliary signals efine as follows sin cos 3 z z x sin ỹ cos z cos sin (6) Journal of Dynamic Systems, Measurement, an Control DECEMBER 000, Vol. Õ 67

3 3 cos sin 3 x sin ỹ cos T. (7) In orer to facilitate the subsequent stability analysis, we utilize the fact that x sin ỹ cos w z z (8) to rewrite 6 an 7 in terms of the auxiliary variables efine in 9 as follows sin cos 3 z z w z z z cos sin (9) 3 cos sin T 3 w z z. (0) 3 Smooth Variable Structure-Like Control Our control objective is to esign a controller for the transforme kinematic moel given in that forces the actual position/orientation of the WMR to track the reference timevarying position/orientation generate in 8. To facilitate the subsequent control evelopment, we efine an auxiliary error signal z (t) R as the ifference between the subsequently esigne auxiliary signal z (t) R an the transforme variable z(t), efine in 9, as follows z z z. () 3. Control Formulation. Base on the open-loop tracking error ynamics given in an the subsequent stability analysis, we esign the auxiliary kinematic control signal u(t) as follows u u a k z () where the auxiliary control signal u a (t) R is efine as u a k w f Jz z, (3) the auxiliary signal z (t) is efine by the following oscillatorlike relationship ż z k w f w Jz z T 0 z 0 0, (4) the auxiliary terms (t) R an (t) R are efine as k w k w f (5) an 0 exp t (6) respectively, f (z,v r,t) was efine in 3, k (t), k (t) R are positive, time-varying functions selecte as follows k k s w c k k s, (7) z c (w,z,t), (w,z,t) R are positive bouning functions, an k s, 0,,, c, c R are positive, constant control gains. Remark. In orer to facilitate the subsequent stability analysis, we note that the auxiliary signals efine in (9) an (0) can be upper boune as follows (8) where the positive bouning functions (w,z,t), (w,z,t) R are efine as follows w z z 3 z z 3 (9) z z 3 w (30) an 4 R is a positive bouning constant selecte as follows 4. (3) Remark 3. Motivation for the structure of (4) is obtaine by taking the time erivative of z T z as follows t z T z z T ż z T z k w f w Jz (3) where 4 has been utilize. After noting that the matrix J of () is skew symmetric, we can rewrite (3) as follows t z T z z T z. (33) As result of the selection of the initial conitions given in (4), it is easy to verify that z T z z (34) is a unique solution to the ifferential equation given in (33). The relationship given by (34) will be use uring the subsequent error system evelopment an stability analysis. Remark 4. Note that the exponential term in (6) is not necessary for the subsequent stability analysis. That is, if 0 is selecte as 0 0 then the subsequent stability proof is still vali. The motivation for selecting 0 0 is to provie the esigner with increase flexibility with regar to ensuring that the control effort is maintaine at a reasonable magnitue. Specifically, for the case of large initial tracking error, the magnitue of the control coul possibly be reuce through the selection of Close-Loop Error System Development. To etermine the close-loop tracking error ynamics for w(t), we substitute for u(t) in the open-loop tracking error system given by a an subtract u T a Jz to the resulting expression, an then utilize to rewrite the ynamics for w(t), as follows ẇ u a T Jz u a T Jz f (35) where the fact that J T J was utilize. Finally, by substituting 3 for only the secon occurrence of u a (t) in 35 an then utilizing the equality given by 34 an the fact that J T J I note that I enotes the stanar ientity matrix, we can obtain the final expression for the close-loop tracking error system for w(t) as follows ẇ u T a Jz k w. (36) To etermine the close-loop error system for z (t), we take the time erivative of, substitute 4 for ż (t), an then substitute for ż(t) to obtain the following expression z8 z k w f w Jz u. (37) After substituting for u(t), an then substituting 3 for u a (t) in the resulting expression, we can rewrite 37 as follows z8 z w Jz z k z. (38) After substituting 5 for only the secon occurrence of (t) in 38 an using the fact that JJ I, we can cancel common terms an rearrange the resulting expression to obtain 68 Õ Vol., DECEMBER 000 Transactions of the ASME

4 z8 k z wj k w f Jz z (39) where has been utilize. Finally, since the brackete term in 39 is equal to u a (t) efine in 3, we can obtain the final expression for the close-loop tracking error system for z (t) as follows z8 k z wju a. (40) 4 Stability Analysis Theorem. The kinematic control law given in () (7) ensures the position an orientation tracking errors efine in (0) are GUUB in the sense that x t, ỹ t, t 0 exp 0 t c c exp t 3 (4) where was efine in (6), c, c were efine in (7) an 0,,, 3, 0, an R are some positive constants. With regar to (4) we note that, c, c can be mae arbitrarily small. Proof: To prove Theorem, we efine a non-negative, scalar function enote by V(t) R as follows V w z Tz. (4) After taking the time erivative of 4, making the appropriate substitutions from 36 an 40, an then cancelling common terms, we obtain the following expression V k w k z Tz w z T (43) where we utilize the fact that J T J. After substituting 7 for k (t) an k (t), we can upperboun V (t) of 43 as follows V k s w k s z w w w c z z z c (44) where 8 was utilize. We can now utilize 4 an the facts that w w w c c z z z c c (45) to upper boun V (t) of 44 as follows V k s V c c. (46) After solving the ifferential inequality given in 46, we obtain the following expression V exp k s t V 0 c c exp k k s t. (47) s Finally, we can utilize 4 to rewrite the inequality given by 47 as t exp k s t 0 c c exp k k s t s (48) where the vector (t) R 3 is efine as w z T T. (49) Base on 48 an 49, it is clear that w(t), z (t) L. After utilizing, 34, an the fact that z (t), (t) L, we can conclue that z(t), z (t) L. From 5 an the fact that w(t), z(t), z (t), z (t) L, it is clear from 9, 0, 8 30 that (t), (t), (w, z, t), (w, z, t) L. Base on these facts, we can now use the assumption that v r (t),v r (t) L, 3, 6, an 34 to show that f (z, v r, t), u a (t), ż (t), (t), u(t) L. Now, in orer to illustrate that the Cartesian position an orientation signals efine in are boune, we utilize the inverse of the transformation given in 9 as follows x sin 0 sin cos ỹ cos 0 z cos sin w z. 0 0 (50) Since z(t) L, it is clear from 0 an 50 that (t), (t) L. Furthermore, from 0, 50, an the fact that w(t), z(t), (t) L, we can conclue that x (t), ỹ(t), x c (t), y c (t) L. We can utilize 4 the assumption that v r (t), v r (t) L, an the fact that (t), u(t), x (t), ỹ(t) L, to show that v(t) L ; therefore, it follows from 5 that (t), ẋ c (t), ẏ c (t) L. Stanar signal chasing arguments can now be employe to conclue that all of the remaining signals in the control an the system remain boune uring close-loop operation. In orer to prove that z(t) efine in 9 is GUUB, we can now apply the triangle inequality to to obtain the following boun for z(t) z z z exp k s t 0 c c exp k k s t s 0 exp t (5) where 6, 34, 48 an 49 have been utilize. Base on 48 5, the result given in 4 can now be irectly obtaine. Remark 5. It is clear that if the control terms c an c are set to zero in (44), then the stability analysis woul be the same as for a variable structure controller. We refer to the control scheme as variable structure-like ue to the close resemblance of the stability analysis to the classic variable structure stability analysis. 5 Setpoint Extension Unlike some of the previously propose tracking controllers see,0,, etc. we have not impose any restrictions on the esire trajectory other than the assumption that v r (t), v r(t), q r (t), an q r(t) L ; hence, the position an orientation tracking problem reuces to the position an orientation regulation problem in a similar manner as illustrate in 3. That is, if the control objective is targete at the regulation problem, the esire position an orientation vector, enote by q r x rc y rc r T R 3 an originally efine in 8, becomes an arbitrary esire constant vector. Base on the fact that q r (t) is now efine as a constant vector, it is straightforwar that v r (t) given in 8, an consequently f (z,v r,t) efine in 3 equal zero. We also note that the auxiliary variable u(t) originally efine in 4, is now efine as follows u T v v Tu (5) where the matrix T was efine in 5. Base on the above simplifications, it is easy to show that the result given by Theorem is vali for the regulation problem as well. Furthermore, we note that the propose kinematic control law given in, 3, 4, 5, 6, an 7 can be slightly moifie as illustrate in Remark 5.5 of 4 to reject parametric uncertainty an boune isturbances in the ynamic moel. Journal of Dynamic Systems, Measurement, an Control DECEMBER 000, Vol. Õ 69

5 6 Uncertain Kinematic Moel Extension Several researchers have examine the setpoint regulation problem for a WMR with a kinematic moel containing parametric uncertainties see 7 9. Specifically, the kinematics of the WMR are given as follows q S q Av (53) where q(t) was efine in S( ) was efine in 3, v(t) v v was efine 4, an A R is efine as follows A * 0 (54) 0 p * where *, p * R are uncertain positive parameters that represent the raius of the wheels an the istance between them. 6. Error System Development. In orer to facilitate the subsequent control esign, we utilize the following global invertible transformation e cos 0 e 0 0 e 3 sin ỹ (55) cos sin 0 x where e (t), e (t), e 3 (t) R are auxiliary tracking error variables, an x rc, y rc, r, originally efine in 8, are now efine as arbitrary esire constants. After taking the time erivative of 55 an using 4, 53 55, we can rewrite the open-loop error ynamics in a more convenient form as follows ė ė p ė 3 p*ve3 *v p *v e. (56) Base on the open-loop error ynamics given in 56 an the subsequent stability analysis, we utilize the following smooth, time-varying controller v v k e 3 e v k e e sin t. (57) After substituting 57 into 56 an performing some algebraic manipulation, we obtain the following close-loop error system p * ė * ė 3 p * p * * ė p * * v e 3 k e e sin t (58) k e 3 e v p * *. Remark 6. Note that the close-loop ynamics for e (t) given in (58) represent a stable linear system subjecte to an aitive isturbance given by the prouct e (t)sin(t). If the aitive isturbance is boune (i.e., if e (t) L ), then it is clear that e (t) L. Furthermore, if the aitive isturbance asymptotically vanishes (i.e., if lim t e (t) 0) then it is clear that lim t e (t) Stability Analysis. Theorem. The smooth, timevarying kinematic control law given in (57) ensures global asymptotic position an orientation regulation in the sense that lim x t,ỹ t, t 0. (59) t Proof: To prove Theorem, we efine a non-negative, scalar function enote by V (t) R as follows V p * p * e * * e 3. (60) After taking the time erivative of 60, making the appropriate substitutions from 58, an then canceling common terms, we obtain the following expression V k e 3. (6) Base on 6 an 60, it is clear that e (t), e 3 (t) L an that e 3 (t) L. Since e (t) L, it is clear from Remark 6 that e (t) L. Base on the fact that e (t), e (t), e 3 (t) L,we can utilize 57 an 58 to prove that v (t), v (t), ė (t), ė (t), ė 3 (t) L. Since ė (t), ė (t), ė 3 (t) L, we can conclue that e (t), e (t), e 3 (t) are uniformly continuous. After taking the time erivative of 57 an utilizing the aforementione facts, we can show that v (t), v (t) L, an hence, v (t), v (t) are uniformly continuous. Base on the facts that e 3 (t) L an that e 3 (t) is uniformly continuous, we can now utilize Barbalat s lemma 5 to prove that lim t e 3 (t) 0. After taking the time erivative of the prouct e (t)e 3 (t) an substituting 58 in the resulting expression, we can conclue that t e e 3 * p * e * v e 3 ė k *e. (6) Since the brackete term in 6 is uniformly continuous i.e., e (t),v (t) are uniformly continuous an lim t e 3 (t) 0, we can utilize an extension of Barbalat s lemma to conclue that lim t t e e 3 0 lim * p * e t * v 0. (63) From the secon limit in 63, it is clear that lim t e (t)v (t) 0. From the facts that lim t e (t)v (t) 0 an lim t e 3 (t) 0, we can utilize 57 to conclue that lim t v (t) 0, an hence, from 58, we can prove that lim t ė (t) 0 an lim t ė 3 (t) 0. To facilitate further analysis, we take the time erivative of the prouct e (t)v (t) an utilize 58 to obtain the following expression t e v e 3 cos t ė v e sin t p *k e v. (64) Since the brackete term in 64 is uniformly continuous, lim t ė (t) 0, an lim t e (t)v (t) 0, we can utilize an extension of Barbalat s lemma to conclue that lim t t e v 0 lim e 3 cos t 0. (65) t From the secon limit in 65, it is clear that lim t e (t) 0. Since we have shown that lim t e (t),e (t),e 3 (t) 0, we can now utilize the inverse of the transformation efine in 55, given as follows x sin 0 cos ỹ cos 0 sin e 3 e (66) 0 0 e to obtain the result given in Simulation Results The control law given in 7 was simulate base on the kinematic moel given in where (t), (t), 3 (t) were selecte in a similar manner as in 4 as follows 0.0H t 0.0H t 4 sin (67) 60 Õ Vol., DECEMBER 000 Transactions of the ASME

6 0.0H t 0.0H t 4 cos (68) 3 0.0H t 0.0H t 4 (69) where H( ) enotes the stanar Heavisie step function. The esire reference linear an angular velocity were selecte as v r m/s respectively, where v r sin x r cos r tan r ra/s, (70) x cr 0 0 m, y cr 0 0 m, r ra (7) see Fig. for the resulting reference time-varying Cartesian position an orientation. The Cartesian position/orientation an the auxiliary signal z (t) were initialize as follows x c 0 m, y c 0 m, (7) r 0 0 ra, z 0 0 T. The control gains that resulte in the best performance are given below k s 0, 0, 0, (73) 0.00, c 0.00, c 0.00 where the bouning terms,, 3, an 4 given in 5 an 3 were selecte as follows 0.5, 0.05, 3 0. an 4.0. (74) The position/orientation tracking error of the COM of the WMR an the associate control inputs are shown in Fig. 3 an Fig. 4, respectively. Utilizing the same control gains an initial conitions, we also emonstrate the effectiveness of the propose controller with regar to the regulation problem. That is, with the reference velocity signals in 70 set to zero an the esire position an orientation setpoint selecte as zero, the propose controller yiels position/orientation regulation errors as shown in Fig. 5 with the associate control inputs given in Fig. 6. Note that by increasing the control terms, c, an c, the chattering effect observe in Fig. 4 an Fig. 6 can be eliminate; however, from 4 it is clear that steay-state position/orientation tracking error will be boune by a larger neighborhoo about the origin. To illustrate this fact, the control parameters, c, an c, were increase until the chattering effect was reuce. The resulting values of the control parameters are given below 0.00, c 0.00, c (75) The resulting position/orientation errors an the associate control torque input are given in Figs. 7 an 8 for the tracking controller an Figs. 9 an 0 for the regulation controller. Simulation results for the kinematic moel given in 53 an 54 are also presente to illustrate the effectiveness of the control law given in 57 where *(t) an *(t) were selecte as follows Fig. 3 *.5, *.5 (76) PositionÕorientation tracking errors Fig. 4 Tracking control input Fig. Desire Cartesian trajectory Fig. 5 PositionÕorientation regulation errors Journal of Dynamic Systems, Measurement, an Control DECEMBER 000, Vol. Õ 6

7 Fig. 6 Regulation control input Fig. 9 PositionÕorientation regulation errors Fig. 7 PositionÕorientation tracking errors Fig. 0 Regulation control input Fig. 8 Tracking control input Fig. PositionÕorientation regulation errors an the Cartesian position/orientation was initialize as follows x cr 0 m, y cr 0 m, r 0 ra. (77) The control gains that resulte in the best performance are given below k 0.75, k 0.5. (78) The resulting position/orientation regulation error an the associate control inputs are shown in Figs. an, respectively. 6 Õ Vol., DECEMBER 000 Transactions of the ASME

8 8 Conclusion Fig. Regulation control input In this paper, we esigne a variable structure-like tracking controller for a mobile robot system subject to boune isturbances in the kinematic moel. Through the use of a Lyapunovbase stability analysis, we have emonstrate that; i : the position an orientation tracking errors exponentially converge to a neighborhoo about zero that can be mae arbitrarily small, an ii the controller provies robustness with regar to boune isturbances in the kinematic moel. In aition, we illustrate that the propose controller can be utilize to regulate the position an orientation of the WMR to an arbitrary esire setpoint. Moreover, since the propose tracking controller is smooth, we note that it can be moifie to inclue the ynamic moel of the WMR to enhance the overall robustness. An aitional extension was also provie to illustrate that the smooth, time-varying controller esigne in can be applie to solve the setpoint regulation problem of a WMR with parametric uncertainties in the kinematic moel. It shoul also be note that in aition to the WMR problem, the propose controllers can be applie to other nonholonomic systems see 6 for example. Finally, simulation results provie verification for the propose controllers. References McCloskey, R., an Murray, R., 997, Exponential Stabilization of Driftless Nonlinear Control Systems Using Homogeneous Feeback, IEEE Trans. Autom. Control, 4, No. 5 pp , May. Samson, C., 997, Control of Chaine Systems Application to Path Following an Time-Varying Point-Stabilization of Mobile Robots, IEEE Trans. Autom. Control, 40, No., Jan pp Canuas e Wit, C., an Khennouf, H., 995, Quasi-Continuous Stabilizing Controllers for Nonholonomic Systems: Design an Robustness Consierations, Proceeings of the 3r European Control Conference, pp Corraini, M. L., Leo, T., an Orlano, G., 999, Robust Stabilization of a Mobile Robot Violating the Nonholonomic Constraint via Quasi-Sliing Moes, Proceeings of the American Control Conference, pp Leroquais, W., an Anrea-Novel, B., 996, Moeling an Control of Wheele Mobile Robots Not Satisfying Ieal Velocity Constraints: The Unicycle Case, Proceeings of the 35th Conference on Decision an Control, pp Anrea-Novel, B., Campion, G., an Bastin, G., 995, Control Of Wheele Mobile Robots Not Satisfying Ieal Constraints: A Singular Perturbation Approach, Int. J. Robust Nonlinear Control, 5, No. 5, pp Hespanha, J., Liberzon, D., an Morse, A., 999, Towars the supervisory control of uncertain nonholonomic systems, Proceeings of the American Control Conference, pp Jiang, Z., 999, Robust Controller Design for Uncertain Nonholonomic Systems, Proceeings of the American Control Conference, pp Jiang, Z., 000, Robust Exponential Regulation of Nonholonomic Systems with Uncertainties, Automatica, 36, No., pp , February. 0 Jiang, Z., an Nijmeijer, H., 997, Tracking Control of Mobile Robots: A Case Stuy in Backstepping, Automatica, 33, No. 7, pp Kanayama, Y., Kimura, Y., Miyazaki, F., an Noguchi, T., 990, A Stable Tracking Control Metho for an Autonomous Mobile Robot, Proceeings of the IEEE International Conference on Robotics an Automation, pp Canuas e Wit, C., an Soralen, O., 99, Exponential Stabilization of Mobile Robots with Nonholonomic Constraints, IEEE Trans. Autom. Control, 37, No., pp Dixon, W. E., Dawson, D. M., Zergeroglu, E., an Zhang, F., 000, Robust Tracking an Regulation Control for Mobile Robots, Int. J. Robust Nonlinear Control: Special Issue on Control of Uneractuate Nonlinear Systems, 0, No. 4, pp Dawson, D. M., Briges, M., an Qu, Z., 995, Nonlinear Control of Robotic Systems for Environmental Waste an Restoration, Prentice Hall, Englewoo Cliffs, NJ. 5 Slotine, J. J. E., an Li, W., 99, Applie Nonlinear Control, Prentice Hall, Englewoo Cliffs, NJ. 6 Bloch, A., Reyhanoglu, M., an McClamroch, N., 99, Control an Stabilization of Nonholonomic Dynamic Systems, IEEE Trans. Autom. Control, 37, No., Nov. Journal of Dynamic Systems, Measurement, an Control DECEMBER 000, Vol. Õ 63