A mixed H 2 =H adaptive tracking control for constrained non-holonomic systems

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1 Available online at Automatica 9 (2) Brief Paper A mixed H 2 =H adaptive tracking control for constrained non-holonomic systems Chung-Shi Tseng a;, Bor-Sen Chen b a Department of Electrical Engineering, Ming Hsin University of Science Technology, Hsin-Fong 41, Taiwan b Department of Electrical Engineering, National Tsing Hua University, Hsin-Chu 4, Taiwan Received 8 October 21; received in revised form 1 December 22; accepted 29 January 2 Abstract In this study, a PID type controller incorporating an adaptive control scheme for the mixed H 2=H tracking performance is developed for constrained non-holonomic mechanical systems under unknown or uncertain plant parameters external disturbances. By virtue of the skew-symmetric property of the non-holonomic mechanical systems an adequate choice of a state variable transformation, sucient conditions are developed for the adaptive mixed H 2=H tracking control problems in terms of a pair of coupled algebraic equations instead of a pair of coupled non-linear dierential equations. The coupled algebraic equations can be solved analytically.? 2 Elsevier Science Ltd. All rights reserved. Keywords: PID controller; Adaptive tracking control; Mixed H 2 =H performance; Non-linear uncertain systems; Non-holonomic constraints 1. Introduction Considerable attention has been paid on studying the motion control of non-holonomic mechanical systems in recent years. It is well-known that in rolling or cutting motions, the kinematic constraint equations are all non-holonomic constraints in the classical sense. The dynamics of such systems is also well understood (Bloch, Reyhanoglu, & McClamroch, 1992; Campion, d Andrea-Novel, & McClamroch, 1991; Chang & Chen, 2). Several results have been published in recent years wherein the motion control design of the non-holonomic mechanical systems has been successfully treated (Bloch et al., 1992; Campion et al., 1991; Sarkar, Yum, & Kumar, 1994). In these designs, the dynamic models were assumed to be perfect, exactly known free of external disturbances. Recent works have also been proposed using switching or time-varying controllers to stabilize the non-holonomic systems (Jiang, 2; Tian & Li, 22). In general, switching or time-varying This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Keum- Shik Hong under the direction of Editor Mituhiko Araki. Corresponding author. addresses: cstseng@must.edu.tw (C.-S. Tseng), bschen@ moti.ee.nthu.edu.tw (B.-S. Chen). controllers are dicult to be realized in practical applications. However, only a few studies have been carried out so far to address the control design for the uncertain non-holonomic mechanical control systems with external disturbances (Chang & Chen, 2). The aim of this study is to nd a PID-type controller incorporating an adaptive control scheme for stabilizing the closed-loop non-holonomic mechanical systems. In addition to stability, tracking performance is also an important issue in control system design. Mixed H 2 =H optimal control for linear systems has been studied (Khargonekar & Rotea, 1991; Limebeer, Anderson, & Hendel, 1994). The H 2 tracking design is related to minimizing the linear quadratic cost of tracking error control input. The H tracking design is related to attenuating the eect of external disturbances. The main purpose of this type of control is to design an H 2 optimal control for the worst-case external disturbance whose eect on system output must be attenuated below a desired value. However, it is dicult to develop a mixed H 2 =H control for the non-holonomic mechanical systems with plant uncertainties. In this situation, an adaptive control scheme is incorporated to compensate plant uncertainties in order to improve tracking performance. In this study, sucient conditions are developed for the adaptive mixed H 2 =H control problem in terms of a pair of coupled non-linear dierential equations. By virtue of 5-198//$ - see front matter? 2 Elsevier Science Ltd. All rights reserved. doi:1.116/s5-198()8-4

2 112 C.-S. Tseng, B.-S. Chen / Automatica 9 (2) the skew-symmetric property of the mechanical systems an adequate choice of a state variable transformation (Johansson, 199), the coupled non-linear dierential equations can be transformed into a pair of coupled algebraic equations. The coupled algebraic equations can be solved analytically. In what follows, x(t) L 2 [;t f ]if (xt (t)x(t)) dt for all t f [; ), x(t) L,if x(t) for all t [; ). 2. General model of constrained non-holonomic mechanical systems The dynamic equations of a general constrained non-holonomic mechanical system can be expressed as (Bloch et al., 1992; Campion et al., 1991; Chang & Chen, 2) M(q)q + C(q; q) q + G(q)=B(q) + J T (q) + d ; (1) where q R n denotes the vector of generalized coordinates; M(q) R n n denotes the mass moment of inertia matrix; C(q; q) q R n denotes the vector of the centripetal Coriolis forces; G(q) R n denotes the vector of gravitational forces; R (n m 6 n) denotes the vector of control inputs; B(q) R n denotes the input matrix; R m denotes the vector of constraint forces; J T (q) R n m denotes the constraint matrix which is assumed to be exactly known bounded; d R n denotes the vector of external disturbances which are assumed to be L L 2 [;t f ], t f [; ). For non-holonomic mechanical systems, the m nonintegrable independent constraints (Bloch et al., 1992; Chang & Chen, 2; d Andrea-Novel, Campion, & Bastin, 1995) J (q) q = (2) are considered modelled for the expression of the kinematic constraints, where J : R n R m n. Therefore, the constraint manifold of the non-holonomic mechanical systems is dened as M non-hol = {(q; q) R n R n : J (q) q =}; () which is (2n m)-dimensional. Let F 1 (q);:::;f n m (q) be a set of smooth linearly independent vector elds in the null space of J (q) such that J (q)f(q)=; (4) where F(q)=[F 1 (q) F 2 (q) F n m (q)]. The constraints (2) (4) imply the existence of an (n m)-vector ż such that q = F(q)ż: (5) Using (5), we can obtain ż from q q as follows: ż =[F T (q)f(q)] 1 F T (q) q: (6) Remark 1. It should be noted that the (n m)-vector ż represents internal state, so that (q; ż) are coordinates for M non-hol sucient to describe the constrained motion. For simplicity of design, the following assumptions are required throughout this study. Assumption 1. There exists an (n m)-vector z, z(q), which depends only on the conguration position q; but not on the velocity q (Chang & Chen, 2; Su & Stepanenko, 1994). Moreover, the matrices F(q), B(q), M(q) G(q) are functions of variable z C(q; q) is a function of both variables z ż. F(q) is exactly known [F T (q)f(q)] 1 [F T (q)b(q)] 1 exist. The existence of [F T (q)b(q)] 1 means that all n m degrees of freedom can be independently actuated (Bloch et al., 1992). Dierentiating (5), we obtain q = F z + Ḟż: (7) Therefore, the dynamic model (1) of the non-holonomic mechanical systems can be expressed as the following reduced form (Campion et al., 1991; Chang & Chen, 2; d Andrea-Novel et al., 1995; Su & Stepanenko, 1994): M(q)F(q)z +[M(q)Ḟ(q)+ C(q; q)f(q)]ż + G(q) = B(q) + J T (q) + d : (8) Premultiplying F T (q) into (8), due to F T (q)j T (q)=; we obtain M F (q)z + C F (q; q)ż + G F (q)=f T (q)(b(q) + d ); (9) where M F (q), F T (q)m(q)f(q); C F (q; q), F T (q) [M(q)Ḟ(q)+C(q; q)f(q)] G F (q), F T (q)g(q). Several fundamental properties are listed as follows (Chang & Chen, 2; Su & Stepanenko, 1994). Property 1. The matrix M F (q) is symmetric positivedenite the matrix Ṁ F (q) 2C F (q; q) is skewsymmetric, i.e., x T [Ṁ F (q) 2C F (q; q)]x =; R n m. Assumption 2. Assume that M F (q)ṙ +C F (q; q)r +G F (q)= Y F (q; q; r; ṙ); where Y F is an n p F matrix of known function, the unknown parameter is a constant p F -dimensional vector r R n m is a vector of smooth functions. The state tracking error is dened as z Ĩ (z z d )dt e, z = z z d R(n m) ; (1) z ż ż d where z d denotes the desired reference trajectory. A ltered linkof tracking error s= 1 z I + 2 z + z R n m is dened,

3 C.-S. Tseng, B.-S. Chen / Automatica 9 (2) where 1 ; 2 are some positive constants to be adequately determined later. Then, the tracking error dynamic equations can be expressed as follows: z ė = 1 z ṡ = f(q; q)e + g(q)f T (q) where [B(q) F (ṙ F ;r F ; q; q)+ d ]; (11) F ( )=M(q)F(q)ṙ F +[M(q)Ḟ(q)+C(q; q)f(q)]r F + G(q); r F =ż d 2 z 1 z I ; I n m n m n m = n m I n m n m ; 1 I n m 2 I n m I n m n m I n m n m f(q; q)= n m n m I n m ; f 1 f 2 f f 1 = 1 M 1 F (q)c F(q; q); f 2 = 2 M 1 F (q)c F(q; q) 1 I n m ; f = M 1 F (q)c F(q; q) 2 I n m ; g(q)= n m n m M 1 F (q) : (12) Remark 2. The control objective is to determine a control law such that, for any (q(); q()) M non-hol ; all variables of the closed-loop system are bounded for all t the tracking errors z z d ż ż d are as small as possible in the presence of external disturbance d. Remark. By the non-holonomic constraint q = F(q)ż Assumption 1, the non-holonomic constraint can be written as q = F(z)ż. Therefore, z z d ż ż d imply q q d = F(z d )ż d. An interesting feature of non-holonomic mechanical systems is their ability to access the entire conguration space (Bloch et al., 1992). In spite of having only (n m) degrees of freedom tracking only z to z d ż to ż d ; it is quite intuitive that the trajectory of non-holonomic mechanical systems can arrive at any conguration through proper path planning (Mukherjee & Anderson, 1994). Such a property is common to non-holonomic mechanical systems can be attributed to the non-integrable nature of their dierential (non-holonomic) constraints. Let us consider the controller as the following form: =[F T (q)b(q)] 1 ( m + a ) =[F T (q)b(q)] 1 ( m + Y F ˆ); (1) where m is a PID type controller, fullling the mixed H 2 =H tracking performance, to be determined later a = Y F ˆ is an adaptive controller. Therefore, by Assumption 2, the tracking dynamics in (11) can be written as ė = f(q; q)e + g(q)[ m + Y F + F T (q) d ]; (14) where = ˆ.. Problem formulation with mixed H 2 =H performance.1. Mixed H 2 =H adaptive control problem for the constrained non-holonomic mechanical systems Given a desired disturbance attenuation level 2 weighting matrices Q 2 (t), Q (t), R 2 (t) R (t), the mixed H 2 =H adaptive control problem for the constrained mechanical systems is said to be solved if there exist optimal control law m, adaptive controller a the worst-case disturbance d such that (Limebeer et al., 1994) J 2 ( m; d) 6 J 2 ( m ; d); m L 2 [;t f ] (15) J ( m; d) J ( m; d ); d L 2 [;t f ]; (16) where J 2 ( m ; d )=e T (t f )Q 2f e(t f )+ T (t f ) 2 (t f ) + (e T (t)q 2 (t)e(t)+ T mr 2 (t) m )dt (17) J ( m ; d )=e T (t f )Q f e(t f )+ T (t f ) (t f ) + (e T (t)q (t)e(t)+ T mr (t) m 2 T d d )dt (18)

4 114 C.-S. Tseng, B.-S. Chen / Automatica 9 (2) for some symmetric positive denite matrices Q 2f, 2, Q f. Remark 4. The physical meaning of H performance is that the eect of d on e must be attenuated below a desired level from the viewpoint of energy, no matter what d is, i.e., the L 2 gain from d to e must be equal to or less than a prescribed value 2. In general, is chosen as a small positive value less than 1 for attenuation of d. The tracking performance is better for the smaller attenuation level, however, larger control eort should be paid. 4. Mixed H 2 =H adaptive tracking control for constrained non-holonomic mechanical systems In this section, sucient conditions are provided for the existence of solution for the mixed H 2 =H adaptive control problems. For the convenience of design, we take R 2 (t) = R(t); R (t) = 2 R(t) 2 = = throughout this paper. Then, we have the following results. Theorem 1. For the non-holonomic mechanical systems in (1), if control input (t) is chosen as (t)=[f T (q)b(q)] 1 ( m + Y F ˆ); (19) where the mixed H 2 =H control law m; the adaptive update lawof ˆ the worst-case disturbance d (t) are dened as follows: m(t)= R 1 (t)g T (q)p 2 (t)e(t); (2) ˆ(t)= 1 Y T F (ṙ F ;r F ; q; q)g T (q)p 2 (t)e(t) (21) [ 1 (F T F) 2 R 1 (t) R 1 (t) 2 R 1 (t) ][ g T ] (q)p (t) g T (q)p 2 (t) = n m (24) with the constraint g T (q)p 2 (t)=g T (q)p (t) (25) the terminal conditions Q 2f =P 2 (t f ) Q f =P (t f ); then the mixed H 2 =H adaptive control problem for the non-holonomic mechanical systems is solved by (19) (21). Proof. See Appendix A. The stability of the adaptive mixed H 2 =H control for the non-holonomic mechanical systems is stated as follows. Theorem 2. For the non-holonomic mechanical systems in (1), if control input (t) is chosen as (19) with the control law m the adaptive update lawof ˆ dened as (2) (21), respectively, where P 2 (t) P (t) are symmetric positive denite solutions of the coupled non-linear differential equations in (2) (24) with the constraint in (25), then the tracking error e(t) the estimation error (t) in the error dynamic system (14) are all bounded. Proof. See Appendix B. Remark 5. The estimated value of ˆ may be arbitrarily large. In order to prevent the adaptive parameters from drifting, projection algorithms can be used to deal with the bounded problems of the estimated value of ˆ (Chang & Chen, 2). d(t)= 2 F(q)g T (q)p (t)e(t); (22) where P 2 (t) P (t) are symmetric positive denite solutions of the following coupled non-linear dierential equations Ṗ 2 (t)+p 2 (t)f(q; q)+f T (q; q)p 2 (t)+q 2 (t) [P (t)g(q);p 2 (t)g(q)] 1 n m (F T [ F) g T ] (q)p 2 (t) 1 (F T F) R 1 (t) g T (q)p 2 2 (t) = n m (2) Ṗ (t)+p (t)f(q; q)+f T (q; q)p (t)+q (t) [P (t)g(q);p 2 (t)g(q)] 5. Solution of coupled non-linear dierential equations Let P 2 (t) P (t) be in the following explicit forms, respectively, K 1 K 2 n m P 2 (t)= T K 2 K n m (26) n m n m M F (q) K 4 K 5 n m P (t)= T K 5 K 6 n m ; (27) n m n m M F (q) where K 1, K, K 4 K 6 are symmetric positive denite constant matrices K 2 K 5 are symmetric constant matrices with K 1 K K 2 K 2 K 4 K 6 K 5 K 5. By the

5 C.-S. Tseng, B.-S. Chen / Automatica 9 (2) skew-symmetric property in Property 1, we obtain e T {Ṗ 2 (t)+p 2 (t)f + f T P 2 (t)}e n m K 1 K 2 = e T K 1 2K 2 K e (28) K 2 K n m e T {Ṗ (t)+p (t)f + f T P (t)}e n m K 4 K 5 = e T K 4 2K 5 K 6 e: (29) K 5 K 6 n m It can also be easily checked that g T P 2 (t)= S T g T P (t)= S T () which satisfy the constraint in (25), where S T = [ n m ; n m ;I n m ]. By the results of (28) (), the coupled non-linear dierential equations (2) (24) in Theorem 1 can be reduced to the following coupled algebraic equations n m K 1 K 2 K 1 2K 2 K K 2 K n m + Q 2(t) 2 T S[R 1 (t) 2 2 (F T F)]S T = n m (1) n m K 4 K 5 K 4 2K 5 K 6 + Q (t) K 5 K 6 n m 2 T S[ 1 2 R 1 (t) 2 (F T F)]S T = n m : (2) For the convenience of design, let R 1 (t)=2[a 2 I n m + 2 (F T (q)f(q))]; = bi n m ; Q 2 = Q Q = diag{q 2 11I n m ;q 2 22I n m ;q 2 I n m }; () where a, b,, q 11, q 22, q are all positive constants. The coupled algebraic equations in (1) (2) can be solved by the following equalities: aq11 1 = ; q = q 1 ; q 11 (4) K 5 = a 2 1 I n m ; 2 = a2 q22 2 ; (5) K 4 = a I n m ; K 6 = a 2 2 I n m ; (6) =2; K 1 = K 4 ; K 2 = K 5 ; K = K 6 : (7) Remark 6. There are no general results ensuring the existence of the solutions for the cross-coupled dierential equations (Freiling, Jank, & Abou-Kil, 1996). In this study, one solution is obtained from the coupled algebraic equations in (1) (2). However, the solution may not be unique. From the above analysis, the control law in (19) can be expressed as the following form: (t)=(f T (q)b(q)) 1 ( m + Y F ˆ); (8) where m(t)= 2a[a 2 I n m + 2 (F T (q)f(q))](q 11 z I + 2q 11 q + q22 2 z + q z) (9) ˆ(t)= a b Y F T (q 11 z I + 6. Simulation example 2q 11 q + q 2 22 z + q z): (4) Consider the control system of a vertical wheel rolling without slipping on a plane surface (Campion et al., 1991; Chang & Chen, 2). The dynamic equations of the vertical wheel are m x = 1 ; my = 2 ; I = 2 ; I = 1 1 r cos() 2 r sin(); where details can be found in Chang Chen (2). The MKS units are used in this example, i.e., x y (m), (rad), I I (Kg m 2 ), 1 2 (N), 1 2 (N m), r (m) m (Kg). The non-holonomic constraints are ẋ =r cos() ẏ =r sin(), where r is the radius of wheel. For simplicity, r = 1 (m) is assumed. Let q, [x; y; ; ] T z(q), [; ] T. The matrix [ ] T r cos() r sin() 1 F(q)= 1 so that the relation q = F(q)ż is satised. The unknown parameter =[(m r 2 + I );I ] T. The nominal parameters of the system are assumed to be m = 1 (Kg), I = 6 (Kg m 2 ) I = 5 (Kg m 2 ). The external disturbances are assumed to be square waves with magnitude ±:25 period =5. The desired reference trajectories are d (t) = sin(t) d (t) = cos(t). The initial conditions z() = ż() =

6 116 C.-S. Tseng, B.-S. Chen / Automatica 9 (2) States The Constraint forces time (sec) Fig. 1. The trajectories of d (rad, solid), (rad, dashdot), d (rad/s, dashed) (rad/s, dotted) time (sec) Fig.. The constranit forces 1 (N, solid) 2 (N, dashdot) States The estimated parameters time (sec) Fig. 2. The trajectories of d (rad, solid), (rad, dashdot), d (rad/s, dashed) (rad/s, dotted). ˆ()=[1; 2] T are assumed. The parameters q 11 =q 22 =q = 1; b=:1 a =:2 are used for the simulation example. Figs. 1 4 show the simulation results for attenuation level =:4. As seen in the simulation results, the desired tracking performance for the mixed H 2 =H adaptive control schemes of the uncertain non-holonomic mechanical systems can be achieved using the proposed methods. 7. Conclusions In this paper, a PID type controller along with an adaptive control scheme for the mixed H 2 =H tracking performance is developed in the non-holonomic mechanical systems under unknown parameters external disturbances. Sucient conditions are developed for the time (sec) Fig. 4. The estimated parameters m r [ 2 + I (Kg m 2, solid) Iˆ (Kg m 2, dashdot). adaptive mixed H 2 =H tracking control problem of the non-holonomic mechanical systems in terms of a pair of coupled non-linear dierential equations. By virtue of the skew-symmetric property of the non-holonomic mechanical systems an adequate choice of the specic form of solution, the coupled non-linear dierential equations can be transformed into a pair of coupled algebraic equations. The coupled algebraic equations can be solved analytically. The proposed methods are simple the PID control gain the adaptive control gain can be obtained systematically. Acknowledgements The authors thankprof. M. Araki, the Associate Editor the reviewers for their constructive comments. This

7 C.-S. Tseng, B.-S. Chen / Automatica 9 (2) workwas supported by National Science Council of the R.O.C. government under Grant NSC E Appendix A Proof of Theorem 1. J 2 ( m ; d ) can be rearranged as follows: J 2 ( m ; d )=e T (t f )Q 2f e(t f )+e T ()P 2 ()e() e T (t f )P 2 (t f )e(t f )+ T () + [e T Q 2 e + T mr m () + d dt (et P 2 (t)e + T )] dt: (A.1) By Q 2f =P 2 (t f ); ˆ(t) in(21) with the fact that (t)= ˆ(t) d in (22), we get J 2 ( m ; d)=e T ()P 2 ()e() + T () () { + e T [ Ṗ 2 + P 2 f + f T P 2 + Q 2 [P g; P 2 g] 1 n m (F T F) 2 1 (F T F) R 1 (t) 2 [ g T ]] P e +[ m + R 1 (t)g T P 2 (t)e] T g T P 2 Then, we conclude that J ( m; d) J ( m; d ); d L 2 [;t f ]: (A.7) This completes the proof. Appendix B Proof of Theorem 2. Let us dene the Lyapunov function for (14) as V (t)=e T P e + T : (B.1) By m in (2), the constraint in (25), ˆ(t) in(21) the dierential equation in (24), we obtain V (t) 6 e T Q e + 2 T d d 6 min (Q ) e d 2 ; (B.2) where min (Q ) denotes the minimal eigenvalue of Q. Whenever e d = min (Q ); V (t) 6. This demonstrates that the error signals e in the error dynamic system (14) are bounded (Khalil, 1996). Finally, from (8), we obtain J T (q) = M(q)F(q)z +[M(q)Ḟ(q)+C(q; q)f(q)]ż + G(q) B(q)(F T (q)b(q)) 1 ( m + a ) d : (B.) Since the right-h side of the above equation is bounded, we can conclude that the constrained force is also bounded. This completes the proof. R(t)[ m + R 1 (t)g T P 2 (t)e] } dt: By (2) the control law in (2), we obtain (A.2) References J 2 ( m; d)=e T ()P 2 ()e() + T () (): (A.) Then, we have J 2 ( m; d) 6 J 2 ( m ; d); m L 2 [;t f ]: (A.4) Similarly, by Q f = P (t f ), the constraint in (25), m in (2), ˆ(t) in(21) the dierential equation in (24), we obtain J ( m; d )=e T ()P 2 ()e() + T () () {[ d 1 F T g T P (t)e] T [ d 1 F T g T P (t)e]} dt: (A.5) By d in (22), we obtain J ( m; d)=e T ()P 2 ()e() + T () (): (A.6) Bloch, A. M., Reyhanoglu, M., & McClamroch, N. H. (1992). Control stabilization of nonholonomic dynamic systems. IEEE Transactions on Automatic Control, 7, Campion, G., d Andrea-Novel, B., & McClamroch, N. H. (1991). Controllability state feedbackstabilizability of non-holonomic mechanical systems. In C. Canudas (Ed.), Advanced robot control (pp ). New York: Springer. Chang, Y. C., & Chen, B. S. (2). Robust tracking designs for both holonomic nonholonomic constrained mechanical systems: Adaptive fuzzy approach. IEEE Transactions on Fuzzy Systems, 8(1), d Andrea-Novel, B., Campion, G., & Bastin, G. (1995). Control of wheeled mobile robots not satisfying ideal velocity constraints: A singular perturbation approach. International Journal of Robust Nonlinear Control, 5, Freiling, G., Jank, G., & Abou-Kil, H. (1996). On global existence of solutions to coupled matrix Riccati equations in closed-loop Nash games. IEEE Transactions on Automatic Control, 41(2), Jiang, Z.-P. (2). Robust exponential regulation of nonholonomic systems with uncertainties. Automatica, 6, Johansson, R. (199). Quadratic optimization of motion coordinate control. IEEE Transactions on Automatic Control, 5,

8 118 C.-S. Tseng, B.-S. Chen / Automatica 9 (2) Khalil, H.K. (1996). Nonlinear systems. Upper Saddle River, NJ: Prentice-Hall, Khargonekar, P. P., & Rotea, M. A. (1991). Mixed H 2 =H control: A convex optimization approach. IEEE Transactions on Automatic Control, 6(7), Limebeer, D. J. N., Anderson, B. D. O., & Hendel, B. (1994). A Nash game approach to mixed H 2 =H control. IEEE Transactions on Automatic Control, 9(1), Mukherjee, R., & Anderson, D. P. (1994). A surface integral approach to the motion planning of nonholonomic systems. ASME Journal of Dynamic Systems, Measurement Control, 16, Sarkar, N., Yum, X., & Kumar, V. (1994). Control of mechanical systems with rolling constraints: Application to dynamic control of mobile robots. International Journal of Robotics Research, 1, Su, C. Y., & Stepanenko, Y. (1994). Robust motion/force control of mechanical systems with classical nonholonomic constraints. IEEE Transactions on Automatic Control, 9, Tian, Y.-P., & Li, S. (22). Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control. Automatica, 8, Bor-Sen Chen received the B.S. degree from Tatung Institute of Technology, Taiwan, in 197, the M.S. degree from National Central University, Taiwan, in 197, the Ph.D. degree from the University of Southern California, Los Angeles, USA, in He was a Lecturer, Associate Professor, Professor at Tatung Institute of Technology from 197 to He is now a Professor at National Tsing Hua University, Hsin-Chu, Taiwan, R.O.C. His current research interests include control signal processing. Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times. He is a Research Fellow of the National Science Council the Chair of the Outsting Scholarship Foundation. Chung-Shi Tseng received the B.S. degree from Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, the M.S. degree from the Department of Electrical Engineering Computer Engineering, University of New Mexico, Albuquerque, NM, USA, the Ph.D. degree in the electrical engineering, National Tsing-Hua University, Hsin-Chu, Taiwan. He is now an Associate Professor at Ming Hsin University of Science Technology, Hsin-Chu, Taiwan. His research interests are in non-linear robust control, adaptive control, fuzzy control, robotics.