TRACKING CONTROL OF WHEELED MOBILE ROBOTS WITH A SINGLE STEERING INPUT Control Using Reference Time-Scaling

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1 TRACKING CONTROL OF WHEELED MOBILE ROBOTS WITH A SINGLE STEERING INPUT Control Using Reference Time-Scaling Bálint Kiss and Emese Szádeczky-Kardoss Department of Control Engineering and Information Technology Budapest University of Technology and Economics Magyar Tudósok krt., Budapest, Hungary bkiss@iit.bme.hu, kardoss@seeger.iit.bme.hu Keywords: Abstract: Time-scaling, wheeled mobile robot, flatness, motion planning, tracking control. This paper presents a time-scaling based control strategy of the kinematic model of wheeled mobile robots with one input which is the steering angle. The longitudinal velocity of the mobile robot cannot be influenced by the controller but can be measured. Using an on-line time-scaling, driven by the longitudinal velocity of the robot and its time derivatives, one can achieve eponential tracking of any sufficiently smooth reference trajectory with non-vanishing velocity. The price to pay is the modification of the traveling time along the reference trajectory according to the time-scaling. The measurement of the time derivatives of the velocity is no longer necessary if the tracking controller is designed to the linearized tracking error dynamics. INTRODUCTION The kinematic model of a wheeled mobile robot (WMR) has generally two inputs namely the longitudinal velocity of the rear ais midpoint and the steering angle of the front wheels. Several strategies are applied to control such WMRs with these two inputs including the tracking error transformation based control reported by (Dion et al., ), the sliding mode controller based solution proposed by (Benalia et al., 3), and the behavior based control strategy studied by (Gu and Hu, ). An important property of the model is its differential flatness (Fliess et al., 995; Fliess et al., 999) implying its dynamic feedback linearizability (with a singularity at zero velocities). However, situations may occur where the longitudinal velocity of the WMR is not generated by a feedback controller, but by an eternal source. A practical eample of this scenario is the tracking problem related to a passenger car without automatic gear. In such a situation the human driver needs to generate the velocity of the car with an appropriate management of the gas, clutch, and break pedals while the tracking controller may influence only the angle of the steered wheels. The kinematic model obtained in such a situation is no longer differentially, but orbitally flat (Respondek, 998; Guay, 999), since one of the inputs is lost. The control problem is still the tracking of the reference trajectory but this tracking may become impossible if the velocity of the reference WMR moving along the reference path is always superior to the real velocity generated by the driver. The opposite is also possible such that the velocity of the reference WMR is always inferior to the real velocity generated by the driver. Nevertheless it is epected that the path of the controlled WMR joins the path of the reference WMR for any velocity profile generated by the driver. To achieve eponential tracking in both cases, this paper suggests a time-scaling of the reference path. This time-scaling uses the measurement of the velocity generated by the driver and eventually its time derivatives. A practical mean to obtain these measurements is the use of the ABS signals available on the CAN bus of the vehicle or the use of alternative sensors (e.g. accelerometers). Recall that time-scaling is a commonly used concept to find optimal trajectories, to cope with input saturation, to reduce tracking errors, and to establish equivalence classes of dynamical systems. One may use off-line time-scaling methods to find the time optimal trajectories for robot manipulators (Hollerbach, 984) or for autonomous mobile vehicles (Cuesta and Ollero, 5). The problem with these off-line methods is that no sufficient control input margins are always assured for the closed loop 86

2 TRACKING CONTROL OF WHEELED MOBILE ROBOTS WITH A SINGLE STEERING INPUT - Control Using Reference Time-Scaling control during the tracking. Other algorithms use therefore on-line trajectory time-scaling for robotic manipulators to change the actuator boundaries such that sufficient margin is left for the feedback controller (Dahl and Nielsen, 99). Another concept is to use the tracking error instead of the input bounds in order to modify the time-scaling of the reference path (Lévine, 4; Szádeczky-Kardoss and Kiss, 6). These methods change the traveling time of the reference path according to the actual tracking error by decelerating if the movement is not accurate enough and by accelerating if the errors are small or vanish. Time-scaling is also introduced related to the notion of orbital flatness defined in (Fliess et al., 999) where a Lie-Bäcklund equivalence of dynamical systems is established such that the transformation involved may change the time according to which the systems evolve. Another approach that relates timescaling to feedback linearization is reported in (Sampei and Furuta, 986). The remaining part of the paper is organized as follows. The net section presents the kinematic models of the WMRs. Section 3 studies briefly the flatness properties of the models. Section 4 presents a simple motion planning method. The time-scaling concept is introduced in Section 5. Two tracking feedback laws, both using time-scaling are reported in Section 6. Simulation results are presented in Section 7 and a short conclusion terminates the paper. KINEMATIC WMR MODELS Let us introduce some notations first. Figure depicts a WMR in the y horizontal plane. Let us suppose that the Ackermann steering assumptions hold true, hence all wheels turn around the same point (denoted by P) which lies on the line of the rear ale. It follows that the kinematics of the robot can be fully described by the kinematics of a bicycle fitted to the longitudinal symmetry ais of the vehicle (see Figure ). The coordinates of the rear ale midpoint are given by and y. The orientation of the car with respect to the ais is denoted by the angle θ, hence the WMR evolves on the configuration manifold R S = SE(). The angle of the front wheel of the bicycle with respect to the longitudinal symmetry ais of the robot is denoted by ϕ. We consider u = ϕ as an input. The longitudinal velocity of the rear ale midpoint is denoted by u if it is a control input (two input case) and by v car if not (one input case). The distance l between the front and rear ales equals to one. Then the model equations can be ob- y l (, y) Figure : Notations of the kinematic model. tained after some elementary considerations which result (see also for eample (Rouchon et al., 993; Cuesta and Ollero, 5)) ẋ = u cosθ () ẏ = u sinθ () θ = u tanu. (3) Since time-scaling is involved in the sequel, we precise that the time in this system is denoted by t and ẋ denotes the time derivative of the function (t) such that ṫ =. Consider now the case where the longitudinal velocity is not a control input but an eternal signal v car which is generated by the driver or by any other mean such that the controller has no influence on its evolution. The corresponding model with one input is defined by the equations ẋ = v car cosθ (4) ẏ = v car sinθ (5) θ = v car tanu. (6) Consider now a time different from the time t, denoted by τ. Based on ()-(3), let us define a model evolving with the time τ and given by the equations τ = u τ, cosθ τ (7) y τ = u τ, sinθ τ (8) θ τ = u τ, tanu τ, (9) where the subscript τ denotes the dependence on the time τ and τ is the derivative of the function τ (τ) with respect to τ. (No subscript is used for variables dependent on time t ecept cases where the distinction is necessary). It is obvious that τ = as ṫ =. 3 FLATNESS OF THE MODELS One can easily verify or check in the literature (Rouchon et al., 993; Fliess et al., 995) that the P 87

3 ICINCO 7 - International Conference on Informatics in Control, Automation and Robotics model ()-(3) (respectively (7)-(9)) is differentially flat, hence it can be linearized by a dynamic feedback and a coordinate transformation. The flat output is given by and y (respectively τ and y τ ). The differential flatness property of the models can be eploited both for motion planning and tracking purposes. Given a reference trajectory re f (t) and y re f (t) for the flat output variables and y, which are at least two times differentiable with respect to the time t, one can determine the time functions of θ re f, u,re f, and u,re f which satisfy ()-(3) according to a mapping { re f,...,ẍ re f,y re f,...,ÿ re f } {θ re f,u,re f,u,re f }. The same holds true for the model (7)-(9) evolving with the time τ hence there eists a mapping { τ,re f,..., τ,re f,y τ,re f,...,y τ,re f } {θ τ,re f,u τ,,re f,u τ,,re f }. () The model (4)-(6) is not differentially, but orbitally flat for v car as shown by (Guay, 999). 4 MOTION PLANNING The motion planning is done for the system (7)-(9) eploiting its differential flatness property. The motion planning realizes the mappings τ { τ,re f, τ,re f, τ,re f } () τ {y τ,re f,y τ,re f,y τ,re f } () for τ [,T] where T is the desired traveling time along the path such that the mapping () allows then to calculate the time functions of the remaining variables of the model. Several motion planning schemes can be used to realize () and (). One may want to solve an obstacle avoidance problem in parallel with the generation of the references (Cuesta and Ollero, 5). For the sake of simplicity, seventh degree polynomial trajectories are considered in this paper such that τ,re f = 7 i= a,i τ i, y τ,re f = 7 i= a y,i τ i. (3) The coefficients are obtained as solutions of a set of linear algebraic equations determined by the constraints that the polynomials and their three successive derivatives must satisfy at τ = and τ = T. Notice that the non-zero constraints are no longer respected in the scaled time t for the derivatives of the references unless τ. It follows in particular that the constraints imposed on the longitudinal velocities at τ = (respectively at τ = T ) will be scaled by τ() (respectively by τ(t(t)). The motion planning can be done off-line prior to the tracking and the time-scaling does not need the redesign of the reference. It follows that more involved methods can be also applied including the one involving continuous curvature pathes with Fresnel integrals (Fraichard and Scheuer, 4). 5 TIME-SCALING A time-scaling law, which is defined by the mapping t τ(t) or by its inverse τ t(τ) can be obtained based on the following consideration. Rewrite (4)-(6) as d = v car dt cosθ (4) dy = v car dt sinθ (5) dθ = v car dt tanu. (6) Similarly, rewrite also (7)-(9) as d τ = u τ, dτcosθ τ (7) dy τ = u τ, dτsinθ τ (8) dθ τ = u τ, dτtanu τ,. (9) Consider now the model equations and the following relations obtained from (4)-(6) and (7)-(9) for the inputs of the models dt dτ = t = u τ, v car u = u τ,. () This allows to determine a unique trajectory of the system (4)-(6) for a trajectory of the system (7)-(9) if the time function v car (t) and the initial conditions are given, and one supposes non-vanishing velocity functions v car and u τ,. As far as the initial conditions are considered one may, for instance, suppose that () = τ (), y() = y τ (), θ() = θ τ (). The relations in the other direction are similar and read dτ dt = τ = v car u τ, = u. () u τ, The following proposition summarizes the properties of the time-scaling for trajectories with strictly positive (respectively negative) velocities. Proposition Suppose that one considers trajectories of the different models of the kinematic car such that the velocities v car and u τ, are both strictly positive (respectively strictly negative). Then the timescaling t τ(t) and τ t(τ) defined by ()-() satisfying t() = τ() are such that the functions τ(t) and t(τ) are strictly increasing functions of their arguments (the scaled time never rewinds). 88

4 TRACKING CONTROL OF WHEELED MOBILE ROBOTS WITH A SINGLE STEERING INPUT - Control Using Reference Time-Scaling This property is a general requirement for meaningful time-scaling and it is satisfied both for forward and backward motions of the car. The time-scaling has a singularity if the car is in idle position. Note that for a fied velocity time function v car (t), the time-scaling can be influenced by u τ,, one of the inputs of the flat model evolving with τ. Observe moreover that for fied v car these relations do not define a one-to-one correspondence between the sets of trajectories of the respective systems and the number of inputs is not preserved, hence they are not Lie- Bäcklund isomorphisms (Fliess et al., 999). Suppose that the references are obtained for (7)- (9) and one disposes of the time functions τ,re f, τ,re f, τ,re f, y τ,re f, y τ,re f, y τ,re f. (A simple method is given in preceding section for the planning of the reference motion.) The time-scaling is defined by the mappings { τ,re f,..., τ,re f } { re f (t),...,ẍ re f (t)} () {y τ,re f,...,y τ,re f } {y re f (t),...,ÿ re f (t)} (3) which can be determined using () since t v car τ(t) = dϑ τ() = t() = (4) u ϑ, v car = τu t, (5) v car = τu t, + τ u t, (6) allow to epress τ(t), τ, and τ. Then re f (t) = τ,re f (τ(t)) (7) ẋ re f (t) = τ,re f(τ(t)) τ (8) ẍ re f (t) = τ,re f(τ(t)) τ + τ,re f(τ(t)) τ (9) and one obtains similar epressions for the higher order time derivatives and for the mapping (3). Suppose that a reference trajectory is calculated according to the time τ and that one is looking for an open loop control of the real car to follow the geometry of the reference trajectory. Assume moreover that the reference trajectory is calculated based on the initial conditions of the real car. Then the open loop control signal u (t) can be calculated from the reference u τ,,re f using the time-scaling defined by (). Notice however that the real traveling time for a reference trajectory according the time t will be obtained as t(t). If the driver generating v car accelerates with respect to the reference trajectory than T > t(t). If he/she is more careful than the algorithm providing the value for T than T < t(t). 6 TRACKING FEEDBACK DESIGN Two tracking controllers are presented in this section. The first one is based on the flatness property of the model with two inputs and requires the measurement of the car velocity and its two successive time derivatives. Measurement of the acceleration and its time derivative may be prohibitive for real applications. Therefore another tracking feedback is also suggested which is designed for a system obtained by linearizing the tracking error dynamics around the reference trajectory achieving only local stability of the reference trajectory. 6. Flatness-Based Tracking using Time-Scaling The system ()-(3) can be linearized by dynamic feedback in virtue of its differential flatness property. The resulting linear system is two chains of integrators (3) = ω y (3) = ω y. (3) Suppose that one specifies the tracking behavior in terms of the tracking errors e = re f and e y = y y re f such that the differential equations e (3) + k, ë + k, ė + k, e = (3) e (3) y + k y, ë y + k y, ė y + k y, e y = (3) hold true. The coefficients k a,i (a {,y}, i =,,) are design parameters and have to be chosen such that the corresponding characteristic polynomials have all their roots in the left half of the comple plane. These linear differential equations define another (tracking feedback) for (3) ω = (3) re f k,ë k, ė k, e (33) ω y = y (3) re f k y,ë y k y, ė y k y, e y. (34) Consider now the model described by (4)-(6). This single input model is not differentially flat, hence cannot be linearized by feedback. It follows that the flatness property cannot be (directly) used to solve the tracking problem. Let us study the possibility to use the time-scaling defined above to achieve the desired tracking behavior for the non-differentially flat model (4)-(6) with one input. The idea is to use the differentially flat model (7)- (9) to solve the motion planning problem with the time τ. Then one would use a tracking feedback controller designed again for the flat model which produces u τ, and u τ,. The signal u τ, produced by the controller is used to drive the time-scaling of the reference trajectory designed for the time τ according to (). The control loop is illustrated in Figure for the model (4)-(6) where the controller provides u = ϕ to the single input model. The tracking feedback is 89

5 ICINCO 7 - International Conference on Informatics in Control, Automation and Robotics motion planning ref in time scaling u t, ref in t driver v car tracking controller, y, Figure : Tracking controller with time-scaling. designed using the flatness property of the model with two inputs. Define first the dynamics for the feedback as (, y) car ζ = ζ ζ3 = v (35) ζ = v ϕ = ζ 3 (36) u = ζ (37) where ζ, ζ, and ζ 3 are the (inner) states of the feedback. Observe that ζ and v give precisely the derivatives of u which need to realize the time-scaling in (4)-(6), hence no numerical differentiation is needed. The inputs v and v of the feedback dynamics must be determined such that the tracking errors e (t) and e y (t) satisfy (3) and (3), respectively. For, one needs to determine first, ẋ, ẍ, (3), y, ẏ, ÿ, and y (3) as functions of, y, θ, ζ, ζ, and ζ 3 which are the states of the closed loop system including the measured states of the kinematic car model, and the states of the feedback (35)-(37). After some cumbersome but elementary differentiations one obtains ẋ = ζ cosθ (38) ẍ = ζ cosθ ζ sinθtanζ 3 (39) (3) = v cosθ 3ζ ζ sinθsinζ 3 cosζ 3 ζ3 cosθ ζ3 cosθcos ζ 3 +v ζ sinθ (4) ẏ = ζ sinθ (4) ÿ = ζ sinθ+ζ cosθtanζ 3 (4) y (3) = v sinθ +3ζ ζ cosθsinζ 3 cosζ 3 ζ3 sinθ ζ3 sinθcos ζ 3 v ζ cosθ. (43) These epressions allow to calculate e, ė, ë, e y, ė y, and ë y using the reference trajectory (scaled with t) and the states of the closed loop system. Plugging in these epressions into (3) and (3), and using (3) one gets cosθ sinθ with [ ] A = B ζ sinθ ζ cosθ ] [ [ v ω A = v ω y B 3ζ ζ sinθsinζ 3 cosζ 3 ζ 3 cosθ+ζ3 cosθcos ζ 3 3ζ ζ cosθsinζ 3 cosζ 3 ζ 3 sinθ+ζ3 sinθcos ζ 3 ] (44) (45) where the inverse of the coefficient matri can be calculated symbolically. One obtains [ v [ v ] = cosθ sinθ ζ sinθ cosθ ζ ] [ ] ω A. (46) ω y B The tracking feedback law is defined by (33)-(34), (35)-(37), and by (46). A singularity occurs if ζ = u = which corresponds to zero longitudinal velocity. Another singular situation corresponds to ζ 3 = ϕ = u = ±π/ which may occur if the steered wheels are perpendicular to the longitudinal ais of the car. Singularities imply the loss of controllability of the kinematic car model. 6. Linearized Error Dynamics The above method needed the time derivatives of the velocity to carry out the time-scaling which may be difficult to measure or estimate in real application. The method presented in this section uses a transformation of the tracking error epressed in the configuration variables, and the non-linear model obtained is linearized around the reference trajectory. The linearized model is controlled by a state feedback similar to the one reported in (Dion et al., ). The lost input, which is the longitudinal velocity of the WMR is again replaced by a virtual input which depends on the time-scaling of the reference trajectory. A slightly different kinematic model is used for this method such that the longitudinal velocity of the rear ale midpoint and the tangent of the steering angle (u 3 = tanϕ) are the inputs of the mobile robot. If the one input case is considered, u 3 = tanϕ is the single control input. Suppose, that the desired behavior of the robot is given by the time functions τ,re f (τ), y τ,re f (τ), θ τ,re f (τ) such that these functions identically satisfy (7)-(9) for the corresponding reference input signals u τ,,re f and u τ,3,re f = tanu τ,,re f. We suggest to scale this reference trajectory according to the time t. The scaled reference trajectory is given by re f (t) = τ,re f (τ), y re f (t) = y τ,re f (τ), and 9

6 TRACKING CONTROL OF WHEELED MOBILE ROBOTS WITH A SINGLE STEERING INPUT - Control Using Reference Time-Scaling θ re f (t) = θ τ,re f (τ) and similarly to (7)-(9) τ,re f = u τ,,re f cosθ τ,re f (47) y τ,re f = u τ,,re f sinθ τ,re f (48) θ τ,re f = u τ,,re f u τ,3,re f. (49) The tracking errors are defined for the configuration variables as e = re f, e y = y y re f, and e θ = θ θ re f. Let us now consider the transformation e cosθ sinθ e = sinθ cosθ e e y (5) e 3 of the error vector (e,e y,e θ ) to a frame fied to the car such that the longitudinal ais of the car coincides the transformed ais. Differentiating this equation w.r.t. time t and using the general rule ȧ(τ) = da(τ) a τ τ e θ dt = t = a τ we get the differential equation ė v car u 3 ė = v car u 3 e e + ė 3 e 3 + sine 3 u τ,,re f τ+ w (5) w which describes the evolution of the errors with respect to the reference path. The inputs w and w are w = v car τu τ,,re f cose 3 (5) w = v car u 3 τu τ,,re f u τ,3,re f. (53) Notice that from these inputs w and w the first derivative of the time scaling ( τ) and the real input u 3 = tanϕ can be calculated as τ = v car w u τ,,re f cose 3 (54) u 3 = w + τu τ,,re f u τ,3,re f v car (55) if the reference value for the longitudinal velocity u τ,,re f, the error of the orientation e 3 ±π/, and the longitudinal velocity of the car v car. This system can be linearized along the reference trajectory, i.e. for [ e e e 3 ] T =. The linearized system is controllable if at least one of the reference control inputs (u τ,,re f,u τ,3,re f ) is nonzero. The setpoint of the linearized system obtained from (5) can be locally stabilized by a state feedback of the form [ ] w = K e e w (56) e 3 velocity [m/s].5.5 Longitudinal velocity profiles traveling time of the quick driver time t reference quick driver slow driver traveling time of the slow driver Figure 3: Velocity profiles for the reference, for the quick driver, and for the slow driver. The traveling times are obtained in closed loop. such that the gain matri K puts the eigenvalues of the closed loop system in the left half of the comple plane. The way of calculations is as follows. One suppose that the tracking errors of the configuration variables (e,e y,e θ ) are measured, hence the the error (e,e,e 3 ) can be determined using (5). Then the state feedback (56) allows to calculate w and w. From the actual value of w one can determine τ using the current value of v car, e 3, and the value of u τ,,re f according to the time t obtained by scaling the reference. The input u 3 = tanϕ is calculated according to (55) using w. The function τ(t) is obtained by the on-line integration of τ determined by (54) using the initial condition τ() =. The time distribution of the reference trajectory is finally modified according to τ and τ. Since a linearized model was used for the controller design only local stability is guaranteed. (E.g. if w is not fulfilled, τ in (54) can get a negative value, which is not allowed since time cannot rewind.) 7 SIMULATIONS Eamples are shown to demonstrate the functioning of both time-scaling based tracking controllers for the one input case. 7. Results of Flatness-Based Solution We use the feedback described in Subsection 6., such that u τ, generated by the feedback law drives the time-scaling given in Section 5 together with the measured v car and its two successive time derivatives. The reference trajectory starts from the point ( τ,re f () =,y τ,re f () =,θ τ,re f () = ) and arrives to the point ( τ,re f (T) =,y τ,re f (T) = 3.5,θ τ,re f (T) = ), all 9

7 ICINCO 7 - International Conference on Informatics in Control, Automation and Robotics y Real and reference trajectories Real Reference Figure 4: Real and reference trajectories in the horizontal plane slow driver. time τ Time scaling functions τ(t) slow driver quick driver without scaling 5 5 time t Figure 6: The time-scaling functions τ(t) along the path for the slow and quick drivers. 4 Real and reference trajectories.8 Derivative of τ(t) y Real Reference Figure 5: Real and reference trajectories in the horizontal plane quick driver acceleration of the reference slow driver quick driver without scaling deceleration of the reference 5 5 time t Figure 7: The derivative of the time-scaling functions τ. distances are given in meters and the orientation is given in radians. The traveling time of the reference trajectory is T = 9 seconds. The real initial configuration of the WMR differs from the one used for motion planning, since () =.5, y() =, and θ() = π/4. Two cases are presented such that the geometry of the reference trajectory and the reference velocity profile obtained are the same. The driver s behavior is different for the two cases. In the first case, referred to as the slow driver case, the driver imposes considerably slower velocities than those obtained by the motion planning. In the second case, referred to as the quick driver case, the driver generates higher velocities than the reference velocity profile. All velocity profiles are given in Figure 3. The geometries of the reference trajectories and the real trajectories in the horizontal plane are depicted in Figure 4 (slow driver) and in Figure 5 (quick driver). Eponential tracking of the reference trajectory is achieved for each scenario with similar geometry of the real path. Figure 6 and Figure 7 show the effects of on-line time-scaling. If the car is driven by a slow driver it needed more than 3 seconds accord- ing to time t to achieve the traveling time T = 9 sec which is given for the reference trajectory according to the time τ. The reference in τ was decelerated all along the trajectory ( τ < ). The deceleration is also accentuated at low values of t which corresponds to large tracking errors. The time-scaling is completely different for the quick driver who reaches the end of the trajectory faster according to the time t than according to the time τ which means that the reference was accelerated ecept a short section at the beginning where the tracking error elimination slows down the time-scaling despite the driver s efforts. 7. Results Obtained by State Feedback Here we use the feedback law described in the subsection 6. such that the same reference trajectory and initial configuration were used as in the previous subsection. The reference trajectory and the real path are shown in Figure 8 for the velocity profiles depicted in Figure 9. We achieved eponential tracking. If the difference between the real and reference initial configurations is larger, the linearized model is 9

8 TRACKING CONTROL OF WHEELED MOBILE ROBOTS WITH A SINGLE STEERING INPUT - Control Using Reference Time-Scaling 3.5 Real and reference trajectories REFERENCES y Real Reference Figure 8: Real and reference trajectories in the horizontal plane simulation. velocity [m/s].5.5 Longitudinal velocity profiles Reference Real time t Figure 9: Velocity profiles for the reference and for the car in simulation. no longer valid and the time-scaling may rewind. 8 CONCLUSION The paper presented two time-scaling based tracking control methods for WMRs with one input such that the longitudinal velocity of the vehicle is generated eternally and cannot be considered as a control input. The time-scaling involves the car velocity and its derivatives which need to be measured or estimated. For the tracking controller designed for the linearized error dynamics the time derivatives of the velocity are not needed. The eponential decay of the initial error along the trajectory can be ensured. The results can be etended for the n-trailer case. ACKNOWLEDGEMENTS The research was partially supported by the Hungarian Science Research Fund under grant OTKA T and by the Advanced Vehicles and Vehicle Control Knowledge Center under grant RET 4/4. Benalia, A., Djemai, M., and Barbot, J.-P. (3). Control of the kinematic car using trajectory generation and the high order sliding mode control. In Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, volume 3, pages Cuesta, F. and Ollero, A. (5). Intelligent Mobile Robot Navigation, volume 6 of Springer Tracts in Advanced Robotics. Springer, Heidelberg. Dahl, O. and Nielsen, L. (99). Torque-Limited Path Following by On-Line Trajectory Time Scaling. IEEE Trans. Robot. Automat., 6(5): Dion, W. E., Dawson, D. M., Zergeroglu, E., and Behal, A. (). Nonlinear Control of Wheeled Mobile Robots. In Lecture Notes in Control and Information Sciences. Springer. Fliess, M., Lévine, J., Martin, P., and Rouchon, P. (995). Flatness and Defect of Nonlinear Systems: Introductory Theory and Eamples. Int. J. of Control, 6(6): Fliess, M., Lévine, J., Martin, P., and Rouchon, P. (999). A Lie-Bäcklund Approach to Equivalence and Flatness of Nonlinear Systems. IEEE Trans. Automat. Contr., 44(5): Fraichard, T. and Scheuer, A. (4). From Reeds and Shepps to Continuous-Curvature paths. IEEE Transaction on Robotics and Automation,. Gu, D. and Hu, H. (). Neural Predictive Control for a Car-like Mobile Robot. Robotics and Autonomous Systems, 39: Guay, M. (999). An Algorithm for Orbital Feedback Llinearization of Single-Input Control Affine Systems. Systems and Control Letters, 38:7 8. Hollerbach, J. M. (984). Dynamic Scaling of Manipulator Trajectories. Trans. of the ASME, J. of Dynamic Systems, Measurement, and Control, 6(): 6. Lévine, J. (4). On the Synchronization of a Pair of Independent Windshield Wipers. IEEE Trans. Contr. Syst. Technol., (5): Respondek, W. (998). Orbital Feedback Linerization of Single-Input Nonlinear Control Systems. In Proceedings of the IFAC NOLCOS 98, pages , Enschede, The Netherlands. Rouchon, P., Fliess, M., Lévine, J., and Martin, P. (993). Flatness and Motion Planning: The Car with n- Trailers. In ECC 93, Proceedings of the European Control Conference, pages Sampei, M. and Furuta, K. (986). On Time Scaling for Nonlinear Systems: Application to Linearization. IEEE Transactions on Automatic Control, AC- 3: Szádeczky-Kardoss, E. and Kiss, B. (6). Tracking Error Based On-Line Trajectory Time Scaling. In INES 6, Proc. of th Int. Conf. on Intelligent Engineering Systems, pages