A Path Following Control for Unicycle Robots

A Path Following Control for Unicycle Robots F. Dıaz del R´´ıo,* G. Jimenez, ´J. L. Sevillano, S. Vicente, A. Civit Balcells Robotics and Computer Technology for Rehabilitation Laboratory Facultad de Informatica. ´ Universidad de Sevilla Avda. Reina Mercedes srn. 41012 Sevilla, Spain In this work we present a new path following control for unicycle robots that is applicable for almost all the possible desired paths and whose analysis is very straightforward. First we select the path following method that consists of two steps: choosing a ‘‘projection’’ that relates the actual posture to the desired path and imposing a ‘‘motion exigency’’ to ensure that the robot advances. A ‘‘projection’’ that considers all the error coordinates is selected and closed equations are obtained for it. The uniqueness projection is carefully analyzed and a necessary and sufficient condition is also presented. This condition shows that a slight bound on the curvature derivative of desired paths must be imposed to preserve uniqueness. It is remarkable that the selected path following is applicable for paths containing zero-radius turns, a problem that has never been resolved as far as we know. In addition, an asymptotically stable control law is found using the closed form equation of the proposed path following and the second Lyapunov method. Finally, we show the behavior of the path following and the control law through several simulated and experimental results, using a computerized wheelchair built at our research facility. * To whom all correspondence should be addressed; e-mail: [email protected]. Contract grant sponsor: IASS agreement ‘‘Segundo Convenio de Colaboracion ´ IASS-Universidad de Sevilla.’’ Contract grant sponsor: CICYT. Contract grant number: TER96-2056-C02-01. 1. INTRODUCTION In the last few years there has been a great interest in finding controllers for nonholonomic mobile robots.The peculiar characteristics of the kinematic and dynamic model of these systems make them especially interesting.Moreover there is no doubt that their applications in the next few years will be large, in fields such as intelligent transportation systems, explorer vehicles, and personal or assistant robots. Mobile robots are intrinsically nonlinear systems, because of their kinematic model.Moreover they have more state coordinates than degrees of Ž. freedom DOF , because of their nonholomic conŽ straints except for the special case of omnidirec- . tional robots .Due to this, convergence to a path may acquire a special treatment in mobile robots. Many researchers have studied various tracking methods when the desired trajectory or path is memorized or previously generated.During the last years several methods have been proposed, and in a general sense we can distinguish between two main tracking methods.In a first group we find those that 1᎐6Ž consider time explicitly in the tracking usually . called trajectory tracking , and try to approach the robot to a moving point.In a second one, we find those that do not have timing requirements and try 7᎐14 Ž to converge to a path usually called path follow- . ing .In the latter case the desired path is usually parameterized,15 and the path following is identified with the progress of the descriptor parameter. Usually the parameter to describe a curve is the time, either the recorded time for previous trajectories or the real time when the tracking is done. However, many different parameters are possible. Note that in this case the time dependence is not relevant. Ž. Trajectory tracking TT has been well studied because it is similar to servosystems, and it is guaranteed that the system will converge to the desired trajectory in a deterministic time using an asymptotŽ ically stable control law except for the perturba- . tions that it may suffer .On the other hand, path Ž. following PF is not well suited for systems with strict timing requirements, but it is very suitable for nonholonomic systems and is applicable to many mobile robots since they are not usually involved in hard real-time systems.Although the first approach seems to be the most straightforward, it has been shown that the second is more suitable for many situations in which time is not a critical parameter. This is the case for most applications in mobile industrial robots or assistant robots such as computerized wheelchairs.This situation can be understood if we consider the following example in TT systems: if big perturbations force the system to be at rest, the desired point for trajectory tracking will move unavoidably.This means that errors will grow up to some value that may introduce instability.On the other hand, if PF were used, the desired point will be the same despite these perturbations, because the path’s shape and the real robot state remain the same.This allows the system to overcome large perturbations, avoiding possible unstable states.Thus interest in PF for mobile robots is rapidly growing. Once a tracking method has been chosen, a convergence law must be found.Due to the special characteristics of the mobile robot state equations and the existence of several methods to track the reference, many researchers have investigated various ways to find a stable control law.Perhaps the most frequent contributions are those based on the Lyapunov direct method.2,4,6 There have also been trials to linearize the system using a first-order approximation,10,13 a local time-varying linearization around the equilibrium point,16 or feedback linearization.7,11 Sampei et al.3designed a controller using the exact linearization and time scale transformation.Recently, adaptive and learning control methods have been successfully applied to nonholonomic systems, including mobile robots.17᎐19 Also conversion of these models into chained systems20 ᎐22 has opened other possibilities for finding controllers for these systems.In addition, the techniques above can be mixed to obtain ‘‘hybrid’’ control laws; frequently these laws behave in a different manner according to the proximity to the target.23,24 Finally we can find several excellent compendia in some reports or books.7,25 Our research group has been interested during the last few years in the improvement of electrical wheelchairs,8,26,27 which incorporate advanced features.This field has interested many researchers in Europe during the last decade due to several projects that are trying to improve the quality of life of handicapped people.28,29 Some features that should be incorporated into classical wheelchairs have been stated.In our group we have developed SIRIUS, an advanced wheelchair that includes path recovery of usual trajectories, detection and avoidance of obstacles through simple sensors like sonar, intelligent user interfaces with shared control, etc.Therefore SIRIUS can be considered to be a mobile robot that can be teleoperated too.Discussing these aspects with trainers and users, we have concluded that playing back previously recorded trajectories is a very helpful aid.This avoids the user having to perform the difficult maneuvering of reverse driving and may be very useful in small areas like bathrooms. One of the typical topologies for electric Ž. wheelchairs is the so-called unicycle or 2, 0 -robot, according to the classification made by Campion et al.,30 because their degree of maneuverability is 2 and their number of steering wheels is 0.They include driver motors at each rear wheel that can turn independently forward or backward.Different speeds at each rear wheel cause the turn of the chair.We have studied the complications that this topology introduces in path tracking, and we have discovered that the possibility of paths whose curŽ. vature tends to infinity i.e., a zero-radius turn is a very interesting problem for the path following method. In the next sections we will try to analyze carefully the proposed PF construction and control law. In section 2 we define our robot model and we choose a set of coordinates, which are appropriate for describing a PF approach.In section 3, a PF construction is established based on two suggested steps: choosing a ‘‘projection’’ that relates the actual posture to the desired path, and imposing a ‘‘motion exigency’’ to ensure that robot advances.In section 4 we develop exhaustively the projection for SIRIUS, we find a closed form equation for the descriptor parameter of the desired path, we construct the projection on the absolute coordinate space, and we study the projection uniqueness.Once PF is well stated, we propose an asymptotically stable control law in section 5, which is evaluated through several simulated and real examples in section 6.Finally in section 7 we summarize the conclusions. 2. DEFINITIONS AND ROBOT MODEL Let us consider the mobile robot shown in Fig.1 Ž. whose dimensions are those of SIRIUS and let Ž. t qsX,Y, ␾ be its state coordinates, which repreŽ. sent the coordinates X,Yof a certain point Po Ž. typically the midpoint between the rear wheels on Ž. Ž . the basis of the fixed frame ᑬsO;i,jand the orientation ␾ of the robot with respect to the fixed frame.Other variables that characterize internal states, such as the angles turned by the wheels or Figure 1. Extrinsic and intrinsic robot coordinates. the relative orientation of castor wheels, do not represent any useful state for the tracking problem. The unicycle robot has three state variables but only 2 degrees of freedom as a result of the nonholonomic constraint.We assume that the wheels are nondeformable and that they are moving on a horizontal plane without slip to hold the constraint.Let Ž. t us ␯ , ␻ be the pair of input variables which represent the linear velocity of point Pand the o angular velocity of the robot, respectively. For these vector variables the kinematic model of the unicycle robot can be expressed by the equaŽ. tions that are nonlinear in qand linear in u: cos ␾ 0 Ž. qsBqu;Bs; sin ␾ 0 ˙ 01 Ž. 1 X ␯ Y us;qs ž/ ␻ 0 ␾ In mobile robots the desired trajectory is usually recorded from a previous real trajectory or generated by a path generator module.31 For our purposes both cases are the same, and the term memorized path,reference path, or merely path is used for both of them.For SIRIUS the usual path to be recovered is the previous path done by the user Ž. usually actuating on the joystick , which must be repeated in reverse direction when the user gives this order. A reference or desired path to be followed can be described by a single parameter, namely, r, and it can be expressed as a vector of desired state Ž. Ž Ž. Ž. Ž.. t coordinates qrsXr,Yr, ␾ r. des des des des To study the tracking of a memorized desired Ž. path qrlet us define another intrinsic coordides Ž. Ž Ž. . nate system ᑤsqr;t,nlinked to the path des Ž15 . usually called the Frenet frame ; see Fig.1.tis the unitary vector parallel to the robot orientation in the Ž. desired point qand nthe normal to it.Let e,e des xy be the position errors of point Prelative to these o axis and let ebe the robot orientation error, so ␾ Ž. t ese,e,ewill be the relative errors vector qxy ␾ Žan analogous coordinate system was used by Kanayama et al.,4but their system was linked to the .Ž.ŽŽ.Ž.. t robot itself .Let urs ␯ r, ␻ rbe the des des des desired inputs expressed as a function of the descriptor parameter r. As we are interested in path following, the parameter rwill be chosen through some kind of relation between the actual system’s state qand the memorized path.This relation will give us the desired point qand the way in which parameter r des varies in relation to t,i.e., a state equation for r. This will be obtained in the following sections. Furthermore, applying the chain law for desired inputs, we can get to d ␾ d ␾ dr des des X Ž. Ž. Ž. ␻ tss s ␾ rrs ␻ rr ˙˙ des des des dt dr dt ŽX. where means differentiation with respect to r. The relation for ␯ is analogous if we consider the des ŽŽ.Ž .. length sof the path ␯ tsds rdt .To sum des des des up we can declare that Ž. Ž. utsurr ˙ des des For the chosen frame, error coordinates ecan q be obtained as a rotation around an axis normal to the XOY plane.In fact and according to Fig.1: Ž.Ž . esR ␾ qyq; qdes des Ž. Ž. cos ␾ sin ␾ 0 des des Ž. R ␾ sŽ. Ž. ysin ␾ cos ␾ 0 des des des 0 001 Therefore the general form of state equations for this kind of coordinates is Ž. Ž. Ž. esBeuqBe u 2 ˙qdes qdes q These equations are linear in all the input variables and nonlinear in state variables.In our case, using the above relative variables and coordinates linked to the path, and by simple calculations, the following state equations can be found32 : e ˙e x0 ␻ 0x y ␯ des des e ˙e sq yy ␻ 00y 0des 0 0 0 y ␻ 0 e des e ␾ ˙000 ␾ Ž. cos e0 ␾ ␯ Ž. q3 Ž. sin e0 ž/ ␾ ␻ 0 01 This form agrees with the intuition that error variables must grow with both real and desired posture advancement. Finally, as we are interested in convergence to a Ž path, we will suppose in this work unless other- . wise stated that the desired trajectory has no end. We do not allow the trajectory to end because convergence to a fixed point qcannot be achieved o through a smooth feedback stabilization control law Ž33. a direct result of Brockett’s theorem . 3. PATH FOLLOWING CONSTRUCTION Previous Studies During the last decade there has been a great research effort to develop a tracking based in path following.This has led to several good approaches that emphasized diverse aspects according to the particular characteristics of the analyzed system or the desired paths to be tracked.The most important can be summarized in the following categories: 䢇 2,7,13,20 In a first category the desired point in the path is obtained through a normal projection along the vector that we have called n. Therefore this projection chooses the point of the desired path that has a null ecoordinate x Ž. see Fig.1.Articles in this category impose a constant value for the variable ␯ to guarantee that the system always moves.Finally an asymptotically convergent control law is obtained and behavior for desired paths composed of circles and lines is shown through simulation.Paths containing circles with a Ž small radius we will call turns with a null radius and infinite curvature ‘‘zero-radius . turns’’ are prohibited, so it is ensured that the normal projection exists and is unique. 䢇 34 A similar path following was used in Navlab. In this case, after finding the same normal projection point P, the authors used a quintic a spline to ‘‘connect’’ the actual robot posture with the desired posture on the point that is located at a certain distance Lfar away from P.As Navlab is a car-like robot, it cannot a make zero-radius turns, so these paths were not considered.On the whole, the main drawback for the alternatives that build a fully specified curve between the actual state and the desired path and force the robot to follow this curve26,34 is that the extraction of closed form equations is not easy, and thus the analytical proof of convergence is very difficult. 䢇 10 In a similar approach the desired posture is chosen as the point of the workspace path Ž. closest to the actual position X,Y.This approach has the same restrictions as the previous choice; in fact in the XY plane, this projection coincides with the previous normal projection—a classical result of differential geometry.35 In this work the proposed controller is designed only for straight-line and circular-arc paths to be tracked with a constant velocity. 䢇 3 Another point of view for the projection is to transform the kinematic equations of the mobile robot into a new time scale.In particular, the time scale is chosen to be identical to the distance along the desired path.However the desired paths are limited to straight lines, because the authors are concerned with the tracking of lines and the parking maneuver in a garage.The authors show that the new scale Ž. the distance along the desired path represents the desired posture obtained through the normal projection. 䢇 11 In the last category the projection point chosen by the authors is the one that minimizes the Euclidean distance between the real and Ž. the desired points Psee Fig.1.Using point l Pthey avoid paths with curvature tending to l infinite.Again these authors obtain good conŽ vergence results for several paths circles and . lines through a feedback linearization control law.But this strategy fails when the desired path is a turn around point P.In this case any o Ž actual configuration having different orienta- . tions whose point Pis on the desired posil tion for Pwill have zero distance.That is, the l Ž. couple X,Ydoes not represent the whole ll state of a mobile robot, although these coordinates always change for every trajectory. It is important to mention that previous studies focused on some particular shapes of the desired paths.In opposition, in SIRIUS we must contemplate all the possible desired paths that can be made Ž. by the user usually driving his or her joystick , including zero-radius turns.This is why we use a different path following approach, first proposed in ref.8 and continued in refs.32 and 9. While trajectory tracking construction is eleŽ mentary it requires only choosing the most suitable Ž.. relation rsrt , path following construction is not so simple because it implies some special relationship between the actual point and the global desired path, and between the inputs u.In this work we Ž propose a PF construction based on two steps Fig. . 2.In this figure we begin with a unicycle robot that Ž has three state coordinates three error coordinates . eexpressed relative to the desired path and 2 q Ž. DOF u. First Step: Projection Ž. Ž . The projection fq,rs0orfe,rs0 relates proj proj q Ž real posture with the desired path the dependence on the desired path can be condensed on the de- . scriptor parameter r.This gives us a projecting point on the desired path: it is the desired posture ŽŽ.. qrt at this instant of time.This projection may des Ž. also be expressed as fe,rs0. proj q Figure 2. General path following construction scheme. The projection is also a holonomic constraint between error coordinates; thus it implies the elimination of one state coordinate.Hence the robot posture will now be given by only two error coordiŽ nates plus the parameter rthat provides all the information about the desired posture, e.g., the ref- .Ž. erence frame , instead of the three ese,e,e qxy ␾ Ž. Fig.3.We will call the new error coordinates Ž. 4 Ž. ese,e.Points ethat obey fe,rs0 dep12 qproj q Ž. fine a surface two-dimensional in our case where the robot is placed.Vector ucannot play a role in this step, because we are talking about a geometric projection. Uniqueness of the projected point on the path Ž. qris not always guaranteed, but the inversion des of the projection must be ensured at least locally, so that the convergence can be proved in a neighborhood of the desired path.This problem will be discussed extensively in the next section. A classical example of one of these projections is the normal projection,2,7,20 equivalent to making ex null, i.e., Ž. fe,rs0: es0 proj qx That is, the first error coordinate eis elimix nated and the robot posture is expressed by only Ž.wŽ. two coordinates: ese,ethat are called y, ␪ py ␾ x in these references .The two-dimensional surface is the eaxis extended for all the possible robot orieny tations. It is important to remark that parameter ris Ž now the third state coordinate in trajectory tracking rgives us no state because ris determined only by Figure 3. Coordinate transformation due to the projection. . time , and we should include it to specify the whole Ž. robot posture, now given by r,e,e.In this work 12 we do not consider ras an error coordinate, because it does not play any role in our stabilization probŽ lem it will only be focused on making eª0, p . regardless of r. Ž. Differentiating the projection fe,rs0we proj qŽ obtain the way in which parameter rvaries an . equation for r: ˙ ⭸ f ⭸ f proj proj eqrs0 ˙˙ q ⭸ e ⭸ r q Ž. Now state Eq.2 can be substituted, and solving for r, we finally have ˙ ⭸ fproj Ž. Be ,ru q ⭸ eqŽ. rs4 ˙ ⭸ f ⭸ f proj proj Ž. qBeu des qdes ⭸ r ⭸ eq Ž. If the denominator is null in Eq.4 , the variation of ris undefined.In the next section we will see that this case is equivalent to the non-uniqueness of the chosen projection. A straightforward example of the requation is ˙ the normal projection mentioned before.If we differentiate this projection we get Ž. es0«y ␯ q ␻ es ␯ cos e ˙xdes des y ␾ Ž. Using the chain law for desired input uts des Ž. urr, we finally obtain ˙ des Ž. Ž. Ž . y ␯ rrq ␻ rres ␯ cos e ˙˙ des des y ␾ Ž. ␯ cos e ␾ «rs ˙Ž. Ž. ␯ ry ␻ re des des y This is the same equation for the normal projection, expressed there using the natural arc parameter s, Ž. Ž. Ž. which makes ␯ ss1 and ␻ ssCs, the des des c planar curvature. Second Step: ‘‘Motion Exigency’’ Finally, we need to impose a ‘‘motion exigency’’ to guarantee that the robot moves.Its form and its variation with time depend on the robot topology and even on the application.For example, in car-like Ž robots a simple and adequate exigency usually . found in the literature is ␯ sconstant.This selection is based on the assumption that angular velocities are never big in cars, and thus linear velocity ␯ is ‘‘eliminated’’ and angular velocity is actually the only DOF to converge to a path. However this is not the only suitable motion exigency for mobile robots; for example, another motion exigency used in the literature,3is that obtained for a constant centrifugal force.The resultant exigencies are hyperbolas in the uplane, with a discontinuity in the axis that must be avoided. SIRIUS motors do not have motion restrictions. Thus the angular and linear velocities may be equivalent since both increase linearly with the in- ˙˙ Ž. dependent velocities of the drive wheels ␪ , ␪ , RL according to the expression RR ˙ ␪ 22 R ␯ sD;Ds ž/ ␻ RR ˙ ž/ ␪ Ly 2d2d Furthermore, the desired trajectories made by a user in domestic environments contain indistinctly paths with very low or very high planar curvature Ž. i.e., ␻ r ␯ , due to the narrow areas where des des wheelchairs must move.Then a very adequate motion exigency is a function that splits the whole motion symmetrically between ␯ and ␻ ,i.e., the ellipse Ž. 222 2 Ž. fus0« ␯ qb ␻ sK)05 motexig ␾ mov Ž where Kis the whole motion applied that may mov vary with time or as a function of several factors . according to the specific application .The parameter bis a scale factor to guarantee dimensional homo- ␾ geneity.Besides, it indicates the degree of turning that we want the robot to perform: i.e.,ifbis low ␾ then the robot will turn slowly; however if it is high, the motion exigency should ask the robot for a faster rotation.Therefore Klimits the maximum mov linear velocity and Krbis the maximum angumov ␾ lar velocity. From the practical point of view, slippage avoidance must be imposed to the collected trajectories; i.e., violent movements of SIRIUS cannot be allowed if the user wants to recover a trajectory. Slippage is more likely when angular velocity is high, so this velocity is bounded on paths done by Ž. the user i.e., ␻ , and for the same reason, the real des angular velocity when following a memorized path Ž. i.e., ␻ must also be bounded.This is another role that constant bwill play in the motion exigency. ␾ This bound does not limit the set of feasible desired paths or the convergence to them. Now that we have proposed a PF construction, we will apply it thoroughly to the case of SIRIUS in the next section. 4. ANALYSIS OF THE PATH FOLLOWING PROJECTION FOR SIRIUS Projection Selection Ž. 30 SIRIUS belongs to the group of 2, 0 -robots, one of the most usual mobile robot configurations.Its trajectories are complicated, because it cannot have Ž complete maneuverability that is, it is not omnidi- . rectional , but it can make zero-radius turns.Thus it is a very interesting problem to find a suitable path following projection for these robots.Moreover the uniqueness of the projection must be deeply analyzed to find a condition that ensures its completion. In some applications only a set of state coordiŽ nates is needed for the path following e.g., if only . lines must be followed .In SIRIUS, due to earlier reasons we have to consider the whole robot state. Furthermore we should consider the three coordinates in a similar fashion as long as maneuverability in these robots is very high and they have no additional movement restrictions.Therefore we choose the point in the path nearest to the robot as the projecting point, i.e., the one for which distance is minimal, and a ‘‘good’’ alternative for the distance dis q 222 22 Ž. dseqeqKe 6 qxy ␾␾ where Kis a scale factor to guarantee dimensional ␾ homogeneity.Another advantage of this selection is that it will ensure uniqueness in a tube around a desired path if a slight bound on the desired paths is preserved. Errors in this distance can vary because of two Ž. reasons: real robot movement i.e., because of u Ž. and selection of projecting point i.e., variation or r. To choose the minimal distance point on the desired Ž. path we ‘‘freeze’’ actual robot posture us0 and Ž. ‘‘move’’ along the desired path ris varied looking for the point with a local minimum 2 ⭸ dqXX X 2< s0«2eeq2eeq2Kee s0 us0 xx yy ␾␾␾ ⭸ rus0 Ž. where ⬘holds for differentiation with respect to r ˙ Ž. and with respect to t.Now we must substitute the state equations, expressed with us0, and conŽ. sidering only the variation of rtime is ‘‘frozen’’ , that is, XŽ. Ž. y ␯ rq ␻ re edes des y x XŽ. esy ␻ re ydes x X 0 0 eŽ. ␾ y ␻ r des Substituting in the state equations and simplifying we finally have the projection: Ž. 2Ž. Ž. e ␯ rsyKe ␻ r7 xdes ␾␾ des As our projection function chooses the desired point nearest to the path, its behavior should be intuitive.This can be completely shown32 through several examples.In this article we have selected two simple examples that show the projection behavior in extreme cases, namely, when the desired path is a straight line and when it is a zero-radius Ž. turn Fig.4. Ž. Ž. In the first case ␯ r/0 and ␻ rs0, and des des thus the projection simplifies to es0; i.e., it coinx cides with the classical normal projection.Therefore when the desired path is a ‘‘pure advance,’’ the projection gives the maximum relevance to the coordinate directly linked to the linear velocity, i.e., Ž. e.On the other hand, in the second case ␯ rs0 xdes Ž. and ␻ r/0, and the projection simplifies to des Ž2,7,12,20 es0 normal projection is not defined in this ␾ . case ; i.e., when desired path is a ‘‘pure turn,’’ the maximum relevance is given to the orientation. Another interesting consequence is that projection problems disappear when the path is a circle and the robot is in the center.Normal projection is undefined in this case.Conversely, using the minimal distance projection, the projecting point will be just the desired point with the same orientation or Ž es0 in the center eis constant and eis null for ␾ yx . every desired point .Moreover we could get the normal projection as a particular case, for example, Ž. by taking Ks0 and supposing that ␯ r/0, ␾ des because normal projection to a planar curve coincides with minimal Euclidean distance.35 State Equation for the Descriptor Parameter As we described in the previous section, differentiation of the projection gives us the new state equation for parameter r.Differentiating the projection with respect to time, and substituting the state Ž equations we express explicitly the dependence of the desired inputs, to distinguish between depen- . dence on ror t: Ž. Ž. Ž . Ž. y ␯ tq ␻ teq ␯ cos e ␯ r des des y ␾ des XŽ. qe ␯ rr ˙ xdes 2wŽ.xŽ. 2XŽ. syK ␻ y ␻ t ␻ ryKe ␻ rr ˙ ␾ des des ␾␾ des Figure 4. Projection behavior for a line and for a zero-radius turn. Finally using the chain law, and solving for r: ˙ Ž. Ž . 2Ž. ␯␯ rcos eqK ␻␻ r des ␾␾ des Ž. rs8 ˙XX 2222 Ž. Ž. Ž. Ž. Ž. Ž. ␯ ry ␻ r ␯ reqK ␻ ryKe ␻ rye ␯ r des des des y ␾ des ␾␾ des xdes where we have supposed that the denominator of Ž this equation is not null we will see below that denominator nullity is directly related to non- . uniqueness .Here ␯ and ␻ are the real inputs, but Ž. Ž. ␯ rand ␻ rare the input profiles that dedes des scribe the desired path. We have to remark that this equation seems to contradict the path following construction since it Ž. uses the three error coordinates e,e,e.Neverxy ␾ theless only two of the three errors are independent, and the third one is completely determined by the projection.In the rest of this article we will use the three errors and the parameter rstate equations plus the projection, always bearing in mind that the projection eliminates one of the errors. Previous problems have arisen from the selecŽ. tion of a coordinate system e,e,erelative to the xy ␾ desired path, whose state equations are quite simple.In any case, a coordinate system in which the robot’s state can be represented only by two independent errors can be found.This can be achieved, for example, if we represent the robot trajectories in the three-dimensional configuration space defined Ž. Ž by X,Y, ␭␾ , where the constant ␭ homogeneous . to a distance is introduced to ensure dimensional Ž homogeneity.In this space we can define as usual 15. in differential geometry the Frenet trihedron ᑤ associated to a point in the desired path q.It can des be demonstrated that the projection on the normal Ž plane defined by the normal and binormal vec15. tors will give us a representation with only two Ž. error coordinates.We can name them ese,e, pnb and the actual posture is then expressed by eplus p the chosen parameter r.This procedure is fully detailed in ref.32, and finally we get a very complicated set of state equations.These equations permit us to obtain a driftless representation of the system. Therefore the PF method reduces the system to a driftless system with two errors eand the paramp eter r, according to the method shown in Fig.2: Ž. rsBr,eu ˙rp Ž. 9 Ž. esBr,eu ˙pe p Regardless of their generality, these three-dimensional configuration space equations are not useful in practice mainly for the following reasons: 䢇 A lot of off-line calculations must be done to express the Frenet’s vectors and other magnitudes as a function of the chosen parameter. 䢇 On-line integration of the resultant state equations is far more complex than that resulting Ž. from using the above r,e,e,estate equa- ˙˙ ˙ ˙ xy ␾ tions, despite the fact that the last one needs one additional integration. 䢇 Moreover, the previous calculations may accumulate numeric errors that may be important if complicated magnitudes are involved in the equations. Table I. Different cases for different desired path curvatures. Resultant Possible Case projection Variation of parameter rpair ␯ cos e 1.Line: ␻ s0 ␾ des Ž. es0rse,e ˙ x y ␾ es0Ž. ␯ r xdes 2.‘‘Pure turn’’: ␻ ␻ s0 and Ž. es0rse,e des ˙ ␾ xy Ž. ␻ r des ␯ s0 des 2 3.Circle with Ž. e,e Ž. ␯ cos eqK ␻ Kr y ␾ ␾␾ des 2 curvature: esyKK e rs ˙ x ␾ des ␾ X 22 2 Ž. Ž. Ž. Ž. Ž. Ž. Ž. e,e ␯ ryKr ␯ reqKK r ␯ ryKeK r xy des des des y ␾ des des ␾␾ des Ks ␻ r ␯ des des des Figure 10. Errors recovering a piecewise trajectory in SIRIUS. but in SIRIUS it is not usual to recover a longer path. Another interesting case is when a desired trajectory contains singular points.In the next experiŽ. ment Figs.10 and 11 the user has made a path composed of five pieces: the first, third, and fifth are almost straight lines and the other two are almost Ž zero-radius left turns that will be right turns in the . reverse recovery .When the chair is approaching a singular point, Kis progressively decreased unmov til SIRIUS is sufficiently near to it.Then the new Figure 11. Real and desired paths recovering a piecewise trajectory in SIRIUS. projecting point on the second piece is calculated, and the coordinate change from the first to the Ž. second piece is done as shown in Fig.5.This coordinate change introduces an error discontinuity Ž that will be recovered by the control again some little oscillation remains mainly due to the motor’s . response delay .So the discontinuity’s magnitude depends largely on how near to the singular point the chair is.All of these facts in this change are empirically adjusted to ensure that the chair’s velocity is sufficiently low and the error discontinuity is small.Note that a specific control to approach closer to the singular point is not necessary because this intermediate point is not the user’s goal.The values for constant parameters in these examples are tuned to ensure a faster convergence in SIRIUS: ␶ s0.3s, xy ␶ s0.3s. ␾ 7. CONCLUSIONS We have presented a new path following for unicycle mobile robots and evaluated it in a computerized wheelchair that recovers the paths done by the user in reverse direction.The path following has been designed to be valid for all the possible trajectories.It relates the actual robot posture to the desired path via a geometric projection that considers all the error coordinates.In fact, it chooses the desired path’s point whose distance to the robot is minimal.Therefore the projection can be applied also to desired paths such as straight lines, circles, or zero-radius turns.In the first case the projection will choose the desired point with null longitudinal error, and in the last case it will choose that with null orientation error.Hence the projection emphasizes linear velocity when following a line and angular velocity when approaching to a pure turn.In addition, the uniqueness of the projection is carefully analyzed.As a result, a slight bound is needed Ž for the curvature derivative of desired paths for . bounded errors .Although we have to impose this bound to the collected paths done by the user with his or her joystick, this does not limit the wheelchair’s maneuverability, because the desired paths that are out of the bound are very abrupt. Moreover we impose a ‘‘motion exigency’’ to force the robot to move.This exigency permits the total robot’s movement to be inverted in angular or in linear speed.Finally an asymptotically stable control law is found using the closed form equation of the proposed path following and the second Lyapunov method.The evaluation of the path following and the control law on the wheelchair shows that its behavior is robust under the high perturbations of this system and under high initial errors for any trajectory driven by the user. Ž The authors thank Professor Claude Samson INRIA, . La France for his help and his kind mailing of research reports. 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