initial configuration initial configuration end

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1 Design of trajectory stabilizing feeback for riftless at systems M. FLIESS y J. L EVINE z P. MARTIN x P. ROUCHON { ECC95 Abstract A esign metho for robust stabilization of riftless at systems aroun trajectories is propose. This metho is base on time scaling, which results in controlling the clock. It permits to follow with exponential stability arbitrary smooth trajectories. These trajectories, which may be obtaine from the the motion planning properties of atness, may contain an pass through steay-states. We thus obtain a stabilization esign aroun rest points for many nonholonomic systems. The example of the stanar n-trailer system is treate in etails with simulations for n = 0 an n = 2. Key wors Nonholonomic system, motion planning, trajectory stabilization, atness, mobile robots, time scaling. Introuction After the emonstration in [3] an in [20] of the impossibility of a straightforwar nonlinear extension of the linear stabilizing strategies, Coron [6, 7] has recently shown how to utilize time-varying feeback for stabilizing a very large class of nonlinear plants. The practical esign of such stabilizing laws is now giving rise to a rapily growing literature (see, e.g., [2, 4, 6, 9, 22]). We here attack this problem via a somehow ierent stanpoint. We restrict ourselves to riftless systems which are at. Remember that last property, which is relate to ynamic feeback linearization [5] an is quite often verie in practice [0, 7, 8], permits to tackle in a most ecient way the motion planning problem. We are thus lea to consier stabilization aroun a given trajectory. This question, espite its importance, has perhaps receive less attention (see, nevertheless [22]). We introuce time-scaling, which may be interprete as controlling the clock [9]. This tool, which seems perhaps surprising at a rst glance, permits to avoi some singularities that are the genuine sources of the mathematical an practical iculties of nonlinear stabiliza- This work was partially supporte by the G.R. \Automatique" of the CNRS an by the D.R.E.D. of the \Ministere e l'eucation Nationale". y Laboratoire es Signaux et Systemes, CNRS-Supelec, Plateau e Moulon, 992 Gif-sur-Yvette Ceex, FRANCE. Tel: 33 () iess@lss.supelec.fr z Centre Automatique et Systemes, Ecole es Mines e Paris, 35, rue Saint-Honore, Fontainebleau Ceex, FRANCE. Tel: 33 () levine@cas.ensmp.fr x Centre Automatique et Systemes, Ecole es Mines e Paris, 35, rue Saint-Honore, Fontainebleau Ceex, FRANCE. Tel: 33 () martin@cas.ensmp.fr { Centre Automatique et Systemes, Ecole es Mines e Paris 60, B Saint-Michel, Paris Ceex 06, FRANCE. Tel: 33 () rouchon@cas.ensmp.fr tion. The paper is organize as follows. In section 2, we briey recall what is a at system. Section 3 eals with the general result. In section 4, the control esign for the stanar n-trailer system is sketche. Simulations for the car without trailer an the car with two trailers are presente. 2 Flat systems More etails can be foun in [8, 9, 0]. A control system is sai to be (ierentially) at if the following conitions are satise:. there exists a nite set y = (y ; : : : ; y m ) of variables which are ierentially inepenent, i.e., which are not relate by any ierential equations. 2. the y i 's are ierential functions of the system variables, i.e., are functions of the system variables (state an input) an of a nite number of their erivatives. 3. Any system variable is a ierential function of the y i 's, i.e., is a function of the y i 's an of a nite number of their erivatives. We call y = (y ; : : : ; y m ) a at or linearizing output. Its number of components equals the number of inepenent input channels. For a \classic" ynamics, _x = f(x; u); x = (x ; : : : ; x n ); u = (u ; : : : ; u m ); () atness implies the existence of a vector-value function h such that y = h(x; u ; : : : ; u () ; : : : ; u m ; : : : ; u (m) m );

2 where y = (y ; : : : ; y m ). The components of x an u are, moreover, given without any integration proceure by the vector-value functions A an B: x = A(y ; : : : ; y () ; : : : ; y m ; : : : ; y m (m) ) u = B(y ; : : : ; y (+) ; : : : ; y m ; : : : ; y m (m+) ): (2) The motion planning problem for () consists in ning a control trajectory [0; T ] 3 t! u(t) steering the system from state x = p at t = 0 to the state x = q at t = T. When the system is at, this problem is equivalent to ning a at output trajectory [0; T ] 3 t! y(t) such that p = A(y (0); : : : ; y () (0); : : : ; y m (0); : : : ; y (m) m (0)) an q = A(y (T ); : : : ; y () (T ); : : : ; y m (T ); : : : ; y (m) m (T )): Since the mapping (y ; : : : ; y () ; : : : ; y m ; : : : ; y (m) m )! A(y ; : : : ; y () ; : : : ; y m ; : : : ; y (m) m ) is locally onto, in general, the problem consists in ning a smooth trajectory t! y(t) with prescribe values for some of its erivatives at time 0 an time T an such that [0; T ] 3 t! A(y (t); : : : ; y () (t); : : : ; y m (t); : : : ; y m (m) (t)) non negative, strictly increasing function such that _(0) = _(T ) = 0. an [0; T ] 3 t! B(y (t); : : : ; y (+) (t); : : : ; y m (t); : : : ; y m (m+) (t)) are well ene smooth functions. In [0, 7, 8], we apply this metho an provie a simple solution to the motion planning for systems stuie in [, 2, 4, 5] an escribing the nonholonomic motion of a car with n trailers. We also remark that natural parametrizations instea of time parametrizations of the linearizing output curves fy(t)j tg simplify the calculations (Frenet formula) an bypass singularities in (2) when _y = 0. We signicantly prolonge this iea here: time-scaling is not only important for ecient computation of open-loop steering controls but also can be very useful for the esign of trajectory stabilizing feeback controllers. 3 Trajectory stabilization Consier the at riftless system _x = mx i= u i f i (x); x 2 R n (3) with y i = h i (x), i = ; : : : ; m as at output (the f i 's an h i 's are smooth functions an the vector els f i are linearly inepenent for all x). Then x an u are given by (2) with A an B ene on open an ense subsets of R + : : :R m + an R +2 : : :R m +2, respectively. Consier the change of parametrization t 7! (t) with an increasing function. The system equation (3) remains unchange by replacing t by an u by u _ instea of u. Thus, uner such transformations, the rst equation of (2) giving x remains unchange whereas the secon one, giving u, is multiplie by _. Theorem Consier (3) an assume that there exist (a j i ), i m an j i, such that A((a j i )) = 0 an the map A is a local submersion aroun (a j i ). Then, for all z 2 R n close to 0 an T > 0, there exists a smooth open-loop control [0; T ] 3 t 7! u(t) steering (3) from x(0) = z, u(0) = 0, to x(t ) = 0, u(t ) = 0. There also exists a class of smooth timevarying ynamic feebacks that stabilize the system aroun this reference trajectory in the following sense: the tracking error e(t) 2 R n satises the estimation for t 2 [0; T ], ke(t)k Mke(0)k exp(?(t)=) where e(0) close to 0 an M is inepenent of e(0), > 0 epens on the esign control parameters, [0; t] 3 t 7! (t) epens only on the reference trajectory an is a smooth, Sketch of proof. Since A is onto there exists (b j i ) close to (a j i ) such that z = A((bj i )). There exist S > 0, m smooth functions, [0; S] 3 s 7! y i;c (s) such that j y i;c s j (0) = b j i, j y i;c s j (S) = a j i, an j y i;c s j (s) close to (a j i ) for i = ; : : : ; m, j = ; : : : ; i an s 2 [0; S] (take, e.g., polynomials in s). Take [0; T ] 3 t 7! (t) 2 [0; S] a smooth increasing function such that (0) = 0, (T ) = S an _(0) = _(T ) = 0. then the open-loop control u(t) = _(t) B y ;c ((t)); : : : ; + y ;c s ((t)); : : : + : : : ; y m;c ((t)); : : : ; m+ y m;c s ((t)) m+ steers (3) from x = z, u = 0 at t = 0 to x = 0, u = 0 at t = T. As for the car, the linearizing ynamic feeback is constructe in the s-scale. The metho is borrowe from [2]. It relies on the fact that A is a submersion aroun (a j i ). This leas to a smooth linearizing control with a linear an asymptotically stable error ynamics in the s-time-scale where correspons to the less stable tracking pole. 2

3 y n y x n l n n l x l length s = (t) of C. We just sketch here the main steps of the control esign. Set u i = v i _(t), i = ; 2 where v i are new control variables. Since xn = v cos n cos(? ) cos(? 2) : : : cos(n?? n) yn = v sin n cos(? ) cos(? 2) : : : cos(n?? n) we consier the static feeback (regular since all the angles?, : : :, n?? n belong to ]? =2; =2[) Figure : the stanar n-trailer system. v = v cos(? ) cos(? 2)::: cos( n?? n) v 2 = v 2 : (5) 4 The stanar n-trailer systems We follow the moeling assumptions of [5]. The notations are summarize on gure. A basic moel is the following: _x = u cos _y = u sin _' = u 2 _ = u tan l _ = u sin(? ) l. _ n = u l n cos(? ) cos(? 2 ) : : : : : : cos( n?2? n?) sin( n?? n ) (4) where (x; y; ; ; : : : ; n ) is the state, (u ; u 2 ) is the control an l, l, : : :, l n are positive parameters (lengths). We know from [8] that this system is at with the cartesian coorinate of the last trailer (x n ; y n ) as at output. Following the geometric construction of [7], we consier a smooth curve C ene by the natural parametrization [0; L] 3 s 7! (x c (s); y c (s)) 2 R 2 (s is the arc length). Denote by (s) the oriente curvature of C, c (s) the angle of the oriente tangent to C. For T > 0 we consier a smooth real increasing function [0; T ] 3 t 7! (t) 2 [0; L] such that (0) = 0, (T ) = L an _(0) = _(T ) = 0. In [7] openloop controls [0; T ] 3 t 7! u c (t) steering the system from a conguration to another one are explicitly given. They rely on the Frenet relationships of planar curve. They are base on a global ieomorphism between (x; y; '; ; ; : : : ; n ) an (x c ; y c ; c ; ; : : : ; n s n ), the contact structure at orer n + 2 of the curve C followe by (x n ; y n ) (the angles ',?, : : :, n?? n belonging to ]? =2; =2[). We just apply the previous theorem to linearize an stabilize the error ynamics with respect to the arc Then we introuce the ynamic compensator of orer n + 2 i = i+ i = ; : : : ; n + n+2 = w v = v 2 = w 2 : (6) Then the inversion of (4,5,6) in -scale, with (x n ; y n ) as output an (w ; w 2 ) as input, leas to an invertible 2 2 ecoupling matrix an a regular feeback on X = (x; y; '; ; ; : : : ; n ; ; : : : ; n+2 ), w w 2 = (X ) + (X ) u v (7) that linearize the system ynamics with respect to the -scale: n+3 x n = u; n+3 y n = v: n+3 n+3 Stanar linear asymptotic tracking methos can be use to ensure the exponential convergence of the tracking error (x n? x c ; y n? y c ) to zero in the -scale. Expressing this controller in the t-scale reveals no iculties an yiels a smooth time-varying ynamic feeback. For the backwar motions isplaye on gure 2, one has n = 0, l = : m an the three tracking poles correspon to the following lengths l=3:5, l=3 an l=2:5. Notice that, after a istance of aroun m, e(0) is ivie by two an that, for t = T, e(t ) 0. Such asymptotic stabilization strategy is interesting when the length L of the reference trajectory C is much larger than the car length l: typically, we have here L=l > 3. For the backwar motions isplaye on gure 3, one has n = 2, l = 2 m, l = 3 m, l 2 = 2 m an the ve tracking poles correspon to the following lengths 2 m, :8 m, :6 m, :4 m, :2 m. As for the car, such asymptotic stabilization strategy is interesting when the length L of the reference trajectory C is much larger than the system length l + l + l 2 : typically, we have here L=(l + l + l 2 ) > 2. 3

4 initial configuration initial configuration en flat output begin reference trajectory en flat output [4] C Canuas e Wit an O.J. Sralen. Exponential stabilization of mobile robots with nonholonomic con- begin reference trajectory Figure 2: the asymptotic stabilization of a backwar trajectory for the car. 5 Conclusion We have escribe a stabilization metho vali for riftless at systems whose singularities results from the time parametrization. The extension of such metho for complex singularities is an open question. This strategy may be combine with the robust stabilization metho propose in []. This leas to approximate motion planning for general trailer systems incluing non at ones [3], but close to stanar trailer systems. This metho can also be extene to more general systems than riftless ones. The above time scaling is a particular case of clock control introuce in [9]. References [] M.K. Bennani. Commane non lineaire e vehicules sur roues avec remorques. Technical Report Option e n 'etue en automatique, Ecole Polytechnique, Paris, july 994. [2] A.M. Bloch, N.H. McClamroch, an M. Reyhanoglu. Control an stabilization of nonholonomic ynamic systems. IEEE Trans. Automat. Control, 37:746{ 757, 992. [3] R.W. Brockett. Asymptotic stability an feeback stabilization. In Dierential Geometric Control Theory. Birkhauser, Basel, 983. Figure 3: the asymptotic stabilization of a backwar trajectory for the stanar 2-trailer system. straints. IEEE Trans. Automat. Control, 37:79{ 797, 992. [5] B. Charlet, J. Levine, an R. Marino. On ynamic feeback linearization. Systems Control Letters, 3:43{5, 989. [6] J.M. Coron. Global stabilization for controllable systems without rift. Math. Control Signals Systems, 5:295{32, 992. [7] J.M. Coron. Linearize control systems an applications to smooth stabilization. SIAM J. Control Optimization, 32:358{386, 994. [8] M. Fliess, J. Levine, Ph. Martin, an P. Rouchon. Sur les systemes non lineaires ierentiellement plats. C.R. Aca. Sci. Paris, I{35:69{624, 992. [9] M. Fliess, J. Levine, Ph. Martin, an P. Rouchon. Linearisation par bouclage ynamique et transformations e Lie-Backlun. C.R. Aca. Sci. Paris, I- 37:98{986, 993. [0] M. Fliess, J. Levine, Ph. Martin, an P. Rouchon. Flatness an efect of nonlinear systems: introuctory theory an examples. Internat. J. Control, to appear. [] J.P. Laumon an T. Simeon. Motion planning for a two egrees of freeom mobile with towing. In IEEE International Conf. on Control an Applications, 989. [2] Ph. Martin. An intrinsic conition for regular ecoupling. Systems Control Letters, 20:383{39, 993. [3] Ph. Martin an P. Rouchon. Systems without rift an atness. In Proc. MTNS 93, Regensburg, Germany, August 993. [4] S. Monaco an D. Norman-Cyrot. An introuction to motion planning uner multirate igital control. 4

5 In Proc. 3th IEEE Control Decision Conf.,Tucson, pages 780{785, 992. [5] R.M. Murray an S.S. Sastry. Nonholonomic motion planning: Steering using sinusois. IEEE Trans. Automat. Control, 38:700{76, 993. [6] J.B. Pomet. Explicit esign of time-varying stabilizing control laws for a class of controllable systems without rift. Systems Control Letters, 8:47{58, 992. [7] P. Rouchon, M. Fliess, J. Levine, an Ph. Martin. Flatness an motion planning: the car with n-trailers. In Proc. ECC'93, Groningen, pages 58{522, 993. [8] P. Rouchon, M. Fliess, J. Levine, an Ph. Martin. Flatness, motion planning an trailer systems. In Proc. 32n IEEE Conf. Decision an Control, San Antonio, pages 2700{2705, ecember 993. [9] C. Samson. Time-varying feeback stabilization of carlike wheele mobile robots. Internat. J. Robotics Res., 2:55{64, 993. [20] E.D. Sontag an H.J. Sussmann. Remarks on continuous feeback. In Proc. 9th IEEE Control Decision Conf., Albuquerque, pages 96{92, 980. [2] D. Tilbury, J.P. Laumon, R. Murray, S. Sastry, an G. Walsh. Steering car-like systems with trailers using sinusois. In IEEE International Conf. on Robotics an Automation, pages 993{998, 992. [22] G. Walsh, D. Tilbury, S. Sastry, R. Murray, an J.P. Laumon. Stabilization of trajectories for systems with nonholonomic constraints. IEEE Trans. Automat. Control, 39:26{222,