Finite-time control for robot manipulators

Systems & Control Letters 46 (22) 243 253 www.elsevier.com/locate/sysconle Finite-time control for robot manipulators Yiguang Hong a, Yangsheng Xu b, Jie Huang b; a Institute of Systems Science, Chinese Academy of Sciences, Beijing 18, China b Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Hong Kong Received 11 April 2; received in revised form 15 January 22 Abstract Finite-time control of the robot system is studied through both state feedback and dynamic output feedback control. The eectiveness ofthe proposed approach is illustrated by both theoretical analysis and computer simulation. In addition to oering an alternative approach for improving the design of the robot regulator, this research also extends the study ofthe nite-time control problem from second-order systems to a large class ofhigher order nonlinear systems. c 22 Elsevier Science B.V. All rights reserved. Keywords: Robot control; Finite-time regulation; Stabilization; Nonsmooth control 1. Introduction In this paper, we will study the problem ofnite-time regulation ofrobots which aims at designing state or dynamic output feedback control laws such that the position of the robot can be regulated into a desired position in nite time. Conventionally, most ofthe existing results on regulation ofrobots is achieved asymptotically. There are two common nonlinear approaches for achieving asymptotic regulation, namely, the inverse-dynamics control or computed torque control [1], and the gravity-compensation control [15]. Other related work can also be found in [2,5,16]. The investigation on the nite-time control of robots is motivated by the following two considerations. In terms of application, nite-time controller, as the name suggested, results in a closed-loop system whose position converges in nite time as opposed to asymptotic convergence ofconventional controller. Thus, our approach may oer an alternative for the control ofrobot manipulators. In terms oftheory, nite-time convergence is an important control problem on its own, and has been studied in the contexts ofoptimality and controllability, mostly with discontinuous or open-loop control. Recently, nite-time stabilization via continuous time-invariant feedback has been studied from dierent perspectives. In particular, the state feedback and the output feedback nite-time control laws for a double integrator system were given in [3,4,7] and [1], respectively. This paper will address the nite-time regulation for a class of multi-dimensional systems describing robot manipulators. We will investigate two classes ofnite-time regulators that correspond to the conventional The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region (Project No. CUHK38=96E), and partially from National Natural Science Foundation of China. Corresponding author. E-mail address: jhuang@acae.cuhk.edu.hk (J. Huang). 167-6911/2/$ - see front matter c 22 Elsevier Science B.V. All rights reserved. PII: S 167-6911(2)13-5

244 Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 inverse-dynamics control and gravity-compensation control as mentioned above. Both state feedback and output feedback control laws will be considered. We note here that while it is fairly straightforward to extend the results in [3,1] on double integrator systems to the robot systems controlled by inverse-dynamics nite-time controller, it is technically nontrivial to establish the nite-time regulation ofthe robot systems controlled by the gravity-compensation controller since in the later case, the closed-loop system is not homogeneous and the results in [3,1] are not directly applicable. Thus, a dierent technique has to be worked out to address the stability issue. For this reason, we will pay more attention to the gravity-compensation controller. 2. Preliminaries We begin with the review ofthe concepts ofnite-time stability and stabilization ofnonlinear systems following the treatment in [3,4]. Consider the system = f(); f()=; () = ; R n (1) with f : U R n continuous on an open neighborhood U ofthe origin. Suppose that system (1) possesses unique solutions in forward time for all initial conditions. Denition 1. The equilibrium = ofsystem (1) is (locally) nite-time stable ifit is Lyapunov stable and nite-time convergent in a neighborhood U U ofthe origin. The nite-time convergence means the existence ofa function T : U \{} (; ); such that; U R n ; the solution of(1) denoted by s t ( ) with as the initial condition is dened; and s t ( ) U \{} for t [;T( )); and lim t T() s t ( ) = with s t ( )= for t T( ). When U = R n ; we obtain the concept ofglobal nite-time stability. Remark 1. Clearly; global asymptotic stability and local nite-time stability imply global nite-time stability. This observation will be used in the following analysis of nite-time stability. In addition; nite-time stability implies the uniqueness in forward time of the solution x =. Denition 2. Consider a nonlinear system ofthe form ẋ = f(x; u); f(; )=; x R n ;u R m ; (2) where f : R n m R n is globally dened. The origin of(2) is nite-time stabilizable ifthere is feedback control law ofthe form u = (x) with () = such that the origin ofthe closed-loop system ẋ = f(x; (x)) is a nite-time stable equilibrium. Remark 2. It is known that the nite-time stability only exists in the systems ofthe form (1) with f() non-lipschitz. Thus the control law (x) must make the closed-loop system nonsmooth. Therefore; nite-time control laws are often sought from the class of homogeneous functions (e.g.; [3;4]). Thus; let us introduce some concepts regarding homogeneous functions following the treatment of [8;12;13]. Denition 3. Let (r 1 ;:::;r n ) R n with r i ; i=1;:::;n. Let V : R n R be a continuous function. V is said to be homogeneous ofdegree with respect to (r 1 ;:::;r n ); if; for any given ; V ( r1 1 ;:::; rn n )= V (); R n : (3) Let f() =(f 1 ();:::;f n ()) T be a continuous vector eld. f() is said to be homogeneous ofdegree k R with respect to (r 1 ;:::;r n ) if, for any given, f i ( r1 1 ;:::; rn n )= k+ri f i (); i=1;:::;n; R n : (4) System (1) is said to be homogeneous if f() is homogeneous.

Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 245 Finite-time stability ofa homogeneous system was studied in [4], and some ofthe results in [4] that will be used in this paper is summarized by the following two lemmas. Lemma 1. Suppose that system (1) is homogeneous of degree k. Then the origin of the system is nite-time stable if the origin is asymptotically stable; and k. Lemma 2. The following control law u = l 1 sgn(x 1 ) x 1 1 + l 2 sgn(x 2 ) x 2 2 ; 1 1; 2 = 2 1 ; (5) 1+ 1 nite-time stabilizes the following double integrator system: ẋ 1 = x 2 ; ẋ 2 = u: (6) Remark 3. Clearly; (5) is a homogeneous function of degree 2 with respect to (2=(1 + 1 ); 1); and it is continuous but not smooth. In the limiting case where 1 approaches 1; the control becomes the conventional PD controller; which is a smooth control law; and cannot achieve nite-time stability as described in Remark 2. For this reason; we call (5) a (homogeneous) nonsmooth PD controller. The following result is from [1], which extends the result given in [13] or [8] from asymptotical stability to nite-time stability. Lemma 3. Consider the following system = f()+ f(); ˆ f()=; R n ; where f() is a continuous homogeneous vector eld of degree k with respect to (r 1 ;:::;r n ); and fˆ satises f()=. ˆ Assume = is an asymptotically stable equilibrium of the system = f(). Then = is a locally nite-time stable equilibrium of the system if fˆ lim i ( r1 1 ;:::; rn n ) =; i=1;:::;n; : (7) k+ri In the next section, we need to extend Lemma 2 to a multi-dimensional form that entails the following notation adopted in [7]: sig() =( 1 sgn( 1 );:::; n sgn( n )) T R n ; R n ; : 3. Finite-time feedback In what follows, we will study global nite-time control of robot systems described by M(q)q + C(q; q) q + G(q)=; q R n ; (8) where q is the vector ofgeneralized coordinates and is the vector ofexternal torque representing the control input; M(q) denotes the inertia matrix, C(q; q) q includes the Coriolis and centrifugal forces, and G(q) represents the gravitational force. In typical mechanical systems, M(q), C(q; q), and G(q) are all smooth. Some well-known properties of(8) are in order [5, pp. 61 62]: P1. M(q) is positive and bounded; P2. Ṁ(q) 2C(q; q) is skew-symmetric; P3. G(q) 6 G for some bounded constant G.

246 Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 Denition 4 (Finite time regulation problem). Given a desired position q d R n ; nd a feedback control law u = (q; q) such that the equilibrium ofthe closed-loop system at (q; q)=(q d ; ) is globally nite-time stable. and In this section, we will consider the state feedback control law of the following two basic forms: = G(q)+C(q; q) q M(q)[l 1 sig(q q d ) 1 + l 2 sig( q) 2 ]; (9) = G(q) l 1 sig(q q d ) 1 l 2 sig( q) 2 ; (1) where 1 1; 2 =2 1 =(1 + 1 );l 1, and l 2. Recall that the inverse dynamics asymptotic regulator and the gravity-compensation asymptotic regulator take the following forms (see [5]): and = G(q)+C(q; q) q M(q)[l 1 (q q d )+l 2 q] (11) = G(q) l 1 (q q d ) l 2 q; (12) respectively. Thus, control laws (9) and (1) can be obtained from (11) and (12) by simply replacing the conventional PD controller in (11) and (12) with the nonsmooth PD controller. Theorem 1. The equilibrium (q; q) =(q d ; ) of the closed-loop systems composed of robot system (8) and control law (9) is globally nite-time stable. Proof. Let x 1 = q q d ;x 2 =ẋ 1 = q; and x =(x1 T;xT 2 )T ; then the closed-loop system ofthe robot manipulator under the control law (9) can be rewritten in the following form { ẋ1 = x 2 ; x i =(x i1 ;:::;x in ) T R n ; i=1; 2; ẋ 2 = [l 1 sig(x 1 ) 1 + l 2 sig(x 2 ) 2 ]; x=(x1 T ; x2 T ) T R 2n (13) : First note that the system is a multiple version of2-dimensional system; which is in a similar form to what was studied in [11]; and therefore; we can conclude that the solution ofthe system is unique in forward time by using Proposition 2.2 and its remark in [11]. Next, take a Lyapunov function candidate of the form V (x) =l 1 =(1 + )( n i=1 x 1i 1+1 )+ 1 2 xt 2 x 2. Then, V (x) (13) = l 2 ( n i=1 x 2i 1+2 ) 6. Because V (x) together with (13) implies (x 1 ;x 2 ), the equilibrium x = ofsystem (13) is globally asymptotically stable by LaSalle s invariant set theorem. Moreover, (13) is homogeneous ofnegative degree (k = 2 1 ) with respect to (2=(1 + 1 );:::;2=(1 + 1 ); 1;:::;1). Thus, the conclusion follows from Lemma 1. Remark 4. This lemma can be viewed as a higher-dimensional version oflemma 2 (where n = 1). In fact; we can also give an explicit construction ofthe Lyapunov function. With the methods used in [6] and [9]; we can take the Lyapunov function V = 2+2l(3+1)=2 x 1 (3+1)=2 + 1+ 1 x 2 (3+1)=(1+1) + l x1 T x 2 ; 3+ 1 3+ 2 where l =(l 1 =l 2 ) 1=1 with l 2 suciently large; and then obtain that there is a K such that V 6 KV (2+21)=(3+1). From Theorem 1 in [3]; this inequality leads to the settling-time estimation given by T 6 (3+ 1 )V (x ) (1 1)=(3+1) =K(1 1 ) with x the initial state ofthe system.

Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 247 Next, we will consider the property ofthe closed-loop system ofthe robot (8) under control law (1), whose structure is simpler than that of(9). For this purpose, let x 1 = q q d, x 2 =ẋ 1 = q, and x =(x1 T;xT 2 )T, the state equation ofthe closed-loop system ofthe robot manipulator under the control law (1) is { ẋ1 = x 2 ; ẋ 2 = M(x 1 + q d ) 1 [C(x 1 + q d (14) ;x 2 )x 2 + l 1 sig(x 1 ) 1 + l 2 sig(x 2 ) 2 ]: Clearly, x = is the equilibrium of(14). It can be seen that the closed-loop system under this control law is no longer homogeneous. Therefore, Lemma 1 does not apply directly to this case. We will use Lemma 3 to establish the global nite-time stability for this later case. For this purpose, we rewrite (14) as follows: { ẋ1 = x 2 ; ẋ 2 = M(q d ) 1 (15) [l 1 sig(x 1 ) 1 + l 2 sig(x 2 ) 2 ]+f q d(x 1 ;x 2 ) where f q d(x 1 ;x 2 )= M(x 1 + q d ) 1 C(x 1 + q d ;x 2 )x 2 M(q d ;x 1 )(l 1 sig(x 1 ) 1 + l 2 sig(x 2 ) 2 ); (16) with M(q d ;x 1 )=M(x 1 + q d ) 1 M(q d ) 1 It can be easily veried that the following system: { ẋ1 = x 2 ; ẋ 2 = M(q d ) 1 [l 1 sig(x 1 ) 1 + l 2 sig(x 2 ) 2 ] (17) is homogeneous ofdegree 1 k= 2 1 with respect to (r 11 ;r 12 ;:::;r 1n ;r 21 ;r 22 ;:::;r 2n ) with r 1i = r 1 = 2=(1 + 1 ) and r 2i = r 2 = 1 for i =1;:::;n. Theorem 2. Assume the solutions of both (14) and (17) are unique in forward time; then the origin of (14) is globally nite-time stable. Proof. We will rst show that the equilibrium of(14) is globally asymptotically stable. For this purpose; consider a Lyapunov function ( n ) V (x)= 1 2 xt 2 M(x 1 + q d )x 2 + l 1 x 1i 1+1 : (18) 1+ 1 Then; i=1 ( n ) V (x) (14) = x2 T [l 1 sig(x 1 ) 1 + l 2 sig(x 2 ) 2 ]+l 1 x2 T sig(x 1 ) 1 = l 2 x 2i 1+2 6 : i=1 It can be seen that V (x) (14) together with (14) implies (x 1 ;x 2 ). It follows from LaSalle s theorem that the equilibrium ofthe closed-loop system at the origin is globally asymptotically stable. Nevertheless, since system (14) is not homogeneous, we cannot apply Lemma 1 to conclude the global nite-time stability of(14). Thus, we need to appeal to Lemma 3. Consider the closed-loop system (16), and take a Lyapunov function candidate of the form V (x) =l 1 = (1 + )( n i=1 x 1i 1+1 )+ 1 2 xt 2 M(qd )x 2. Then, V (x) (17) = l 2 ( n i=1 x 2i 1+2 ) 6. With invariant set theorem and Lemma 1, the homogeneity ofnegative degree (k ) ofsystem (17) yields the global nite-time stability ofits equilibrium x =.

248 Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 Next, we will conclude from Lemma 3 that the equilibrium of system (15) is locally nite-time stable by showing that f q d( r1 x 11 ;:::; r2 x 2n ) is higher degree with respect to k+r2 in the sense of(7) where r 1 =2=(1 + 1 ) and r 2 = 1. To this end, rst note that, since M(q d + x 1 ) 1 and C(x 1 + q d ;x 2 ) are smooth, and k, M(q d + r1 x 1 ) 1 C(q d + r1 x 1 ; r2 x 2 ) r2 x 2 lim = M(q d ) 1 C(q d ; )x 2 lim k =: k+r2 Next applying the mean value inequality (Theorem 9.19 of[14]) to each entry of M(q d ; r1 x 1 ) gives Thus M(q d ; r1 x 1 )=M( r1 x 1 + q d ) 1 M(q d ) 1 =O( r1 ): lim M(q d ; r1 x 1 )[l 1 sig( r1 x 1 ) 1 + l 2 sig( r2 x 2 ) 2 ] k+r2 = lim O( (r1 k r2) ) = lim O( 2k )=: Thus for any xed x =(x1 T;xT 2 )T R 2n, f q d( r1 x 11 ;:::; r1 x 1n ; r2 x 21 ;:::; r2 x 2n ) lim =; k+r2 which shows the local nite-time stability ofthe equilibrium ofsystem (15) according to Lemma 3. Finally, invoking Remark 1 completes the proof. Remark 5. The uniqueness ofthe solution in forward time ofeither (15) or (17) is not guaranteed in general because ofthe complexity ofthe closed-loop systems. However; for the class of robot systems with primatic joints as shown in Section 4; using the results of[11]; it is easy to conclude the uniqueness ofthe solution in forward time of both (15) and (17). Remark 6. The above results still hold true when the gains; l 1 L i = diag(l 1 i ;:::;l n i );lj i ; i=1; 2; j =1;:::;n; respectively. and l 2 ; are replaced by diagonal matrices: Remark 7. A variation of state feedback control law (1) is given as follows: = G(q) l 1 sat v (sig(q q d ) 1 ) l 2 sat v (sig( q) 2 ) (19) with 1 1; 1 = 2 =(2 2 );l 1 ; l 2 and sat v () = (sat( 1 );:::;sat( n )) T ; where sat( i )isthe saturation function; i.e.; sat( i )= i if i 6 1; sat( i )=1 if i 1; and sat( i )= 1 if i 1; respectively. This control law is bounded since G 6 G ; and it is desirable in case ofactuator saturation. Remark 8. In practice; it is not easy to obtain accurate measurement ofthe velocity q. Thus; it is more desirable to design a control law that relies on the measurements ofthe position q only. Following our work for the double integrator system in [1]; we consider the following dynamic output feedback control law: = G(q) l 1 sig( 1 ) 1 l 2 sig( 2 ) 2 ; (2) { 1 = 2 k 1 sig(e 1 ) 1 ; 2 = M(q) 1 [ G(q) C(q; 2 ) 2 + ] k 2 sig(e 1 ) 2 ; (21)

Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 249 where 1 1; 2 =2 1 =(1 + 1 ); 1 = 1 ; 2 =2 1 1; and e 1 = 1 x 1 ;e 2 = 2 x 2. Under this control law; the closed loop system can be written as follows: ẋ 1 = x 2 ; ẋ 2 = f 1 (x; e) M(q d ) 1 [l 1 sig(x 1 + e 1 ) 1 + l 2 sig(x 2 + e 2 ) 2 ]; (22) ė 1 = e 2 k 1 sig(e 1 ) 1 ; ė 2 = f 2 (x; e) k 2 sig(e 1 ) 2 ; where and f 1 = M(x 1 + q d ) 1 C(x 1 + q d ;x 2 )x 2 M(q d ;x 1 )[l 1 sig(x 1 + e 1 ) 1 + l 2 sig(x 2 + e 2 ) 2 ] f 2 = M(x 1 + q d ) 1 [C(x 1 + q d ;x 2 )x 2 C(x 1 + q d ;x 2 + e 2 )(x 2 + e 2 )] and M(q d ;x 1 ) is as dened in Theorem 2. Associated with (22) is the following system: ẋ 1 = x 2 ; ẋ 2 = M(q d ) 1 [l 1 sig(x 1 + e 1 ) 1 + l 2 sig(x 2 + e 2 ) 2 ]; ė 1 = e 2 k 1 sig(e 1 ) 1 ; ė 2 = k 2 sig(e 1 ) 2 : (23) It can be easily veried that (23) is homogeneous. Theorem 3. Assume the solutions of (22) and (23) are unique in forward time; then the origin of (22) is locally nite-time stable for any positive constants l 1 ;l 2 ;k 1 ; and k 2. The proofis a combination ofthe proofoftheorem 2 here and the proofofproposition 2 in [1]. We omit the prooffor avoiding redundancy. Moreover, it is possible to show that Theorem 3 still holds if(2) is replaced by = G( 1 ) l 1 sig( 1 ) 1 l 2 sig( 2 ) 2,or = G(q) l 1 sat v (sig( 1 ) 1 ) l 2 sat v (sig( 2 ) 2 ), which leads to a bounded output feedback nite-time regulator for the robot system. 4. Example In the section, we will show the performance of the nite-time controllers using a two-link robotic manipulators moving in a plane by Matlab 5.2. Consider a robot system with two prismatic joints ofthe form (8) with [ ] (m1 + m 2 ) M(q)= ; m 2 C =, and [ ] G(q)=g ; m 2 q 2 where g is the acceleration ofgravity, m 1 =18:8, and m 2 =13:2.

25 Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 1.4 1.2 Responses of joint 1 1.8.6.4.2 = 1 = 1/2 = 1/3 = 1/4.5 1 1.5 2 2.5 3 Time t Fig. 1. Time response of q 1 under nite-time regulators and asymptotic regulator. Our controller takes the following general form: = G(q) (k 1 q 1 q1 d 1 sgn(q 1 q1)+k d 2 q 1 2 sgn( q 1 );k 3 q 2 q2 d 1 sgn(q 2 q2) d + k 4 q 2 2 sgn( q 2 )) T ; (24) where 1 1, 2 =2 1 =(1 + 1 ), and k i ; i=1; 2; 3; 4. It is not dicult to obtain that the solution of the closed loop system is unique in forward time with Proposition 2.2 and its remark in [11]. This controller includes the conventional gravity-compensated asymptotic controller as a limiting case with 1 = 2 = 1. The performance ofthe controller is dened by two sets ofparameters, namely, the PD gains (k 1 ;k 2 ;k 3 ;k 4 ), and an additional parameter 1. For comparison, we select two dierent sets ofpd gains and three dierent 1 s for simulations. First we take k 1 =4; k 2 =3; k 3 =12, and k 4 =1 with 1 = 1 3 ; 1 4 ; 1 5 ; 1, respectively. The step responses ofthe positions ofthe rst link, and second link are shown in Figs. 1 and 2, respectively. In both gures, the star line, solid line, dotted line, and dashed line correspond to the controller with 1 = 1 2 ; 1 3 ; 1 4 and 1, respectively. As expected, the curve corresponding to 1 = 1 (hence 2 =1) is the response by an asymptotic regulator while the other three curves exhibit the behaviors ofnite time convergence. Due to the nite-time convergence property, responses ofall the three nite time controllers are clearly faster than the asymptotic controller. The characterization of the quantitative relation between the settling time and parameter 1 is interesting, and worth further investigation. Figs. 3 and 4 repeat the experiment shown in Figs. 1 and 2 with k 1 = 22; k 2 = 2; k 3 = 8, and k 4 = 7. The conclusion is consistent with the rst experiment. 5. Conclusions This paper has studied the problem ofthe nite-time regulation ofrobot systems. The research has oered an alternative approach for improving the design of the robot regulator, and also solved the nite-time control problem for a large class of nonlinear systems. Also simulation shows that the control law can achieve faster

Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 251 1.4 1.2 1 Responses of joint 2.8.6.4.2 = 1 = 1/2 = 1/3 = 1/4.5 1 1.5 2 2.5 3 Time t Fig. 2. Time response of q 2 under nite-time regulators and asymptotic regulator. 1.4 1.2 Responses of joint 1 1.8.6.4.2 = 1 = 1/2 = 1/3 = 1/4.5 1 1.5 2 2.5 3 Time t Fig. 3. Time response of q 1 under nite-time regulators and asymptotic regulator. response than conventional PD control law. This virtue may be attributed to the extra parameters in the control law. Two issues have not been adequately addressed in this paper, namely, the relation ofvalues ofthe controller parameter to the settling time, and uniqueness problems. As mentioned in Remark 4, the answer to the rst issue relies on the explicit construction ofthe Lyapunov function. It is possible to nd an explicit Lyapunov function for the closed-loop system resulting from the rst controller, but it does not seem to be easy to do so for the closed-loop system resulting from the second controller since the closed-loop system is not

252 Y. Hong et al. / Systems & Control Letters 46 (22) 243 253 1.4 1.2 Responses of joint 2 1.8.6.4.2 = 1 = 1/2 = 1/3 = 1/4.5 1 1.5 2 2.5 3 Time t Fig. 4. Time response of q 2 under nite-time regulators and asymptotic regulator. homogeneous. As for the second issue, we have given a complete analysis ofthe stability property ofthe closed-loop system resulting from the rst controller. However, for the closed-loop system resulting from the second controller, our results rely on the assumption ofthe uniqueness ofthe solution in forward time. The uniqueness condition may be veried for robot systems with prismatic joints, but may not be guaranteed for more general robot systems. Thus this issue remains to be an unsolved and challenging one. Acknowledgements The authors wish to thank anonymous reviewers for many constructive comments about the paper. References [1] A. Aubin, C. Canudas de Wit, H. Sidaoui, Dynamic model simplication ofindustrial manipulators, IFAC Symposium on Robot Control, Vienna, 1991, pp. 9 14. [2] H. Berghuis, H. Nijmeijer, Global regulation ofrobots using only position measurement, Systems Control Lett. 21 (1993) 289 294. [3] S. Bhat, D. Bernstein, Finite-time stability ofhomogeneous systems, Proceedings ofthe Automatic Control Conference, Albuquerque, NM, 1997, pp. 2513 2514. [4] S. Bhat, D. Bernstein, Continuous nite-time stabilization ofthe translational and rotational double integrators, IEEE Trans. Automat. Control 43 (1998) 678 682. [5] C. Canudas de Wit, B. Siciliano, G. Bastin, Theory ofrobot Control, Springer, London, 1996. [6] J. Coron, L. Praly, Adding an integrator for the stabilization problem, Systems Control Lett. 17 (1991) 89 17. [7] V. Haimo, Finite time controllers, SIAM J. Control Optim. 24 (1986) 76 77. [8] H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, in: S. Elaydi (Ed.), Dierential Equations, Stability and Control, Marcel Dekker, New York, 1991, pp. 249 26. [9] Y. Hong, Finite-time stabilization and stabilizability ofa class ofcontrollable systems, preprint. [1] Y. Hong, J. Huang, Y. Xu, On an output feedback nite-time stabilization problem, IEEE Trans. Automat. Control 46 (21) 35 39. [11] M. Kawski, Stabilization ofnonlinear systems in the plane, Systems Control Lett. 12 (1989) 169 175.

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