Automatica 40 (2004) 2111 – 2119 www.elsevier.com/locate/automatica Brief paper Robust adaptive motion/force tracking control design for uncertain constrained robot manipulators夡 Chian-Song Chiu1 , Kuang-Yow Lian∗ , Tsu-Cheng Wu Department of Electrical Engineering, Chung-Yuan Christian University, Chung-Li 32023, Taiwan Received 16 April 2001; received in revised form 12 December 2003; accepted 30 June 2004 Available online 11 September 2004 Abstract In the presence of uncertain constraint and robot model, an adaptive controller with robust motion/force tracking performance for constrained robot manipulators is proposed. First, robust motion and force tracking is considered, where a performance criterion containing disturbance and estimated parameter attenuations is presented. Then the proposed controller utilizes an adaptive scheme and an auxiliary control law to deal with the uncertain environmental constraint, disturbances, and robotic modeling uncertainties. After solving a simple linear matrix inequality for gain conditions, the effect from disturbance and estimated parameter errors to motion/force errors is attenuated to an arbitrary prescribed level. Moreover, if the disturbance and estimated parameter errors are square-integrable, then an asymptotic motion tracking is achieved while the force error is as small as the inversion of control gain. Finally, numerical simulation results for a constrained planar robot illustrate the expected performance. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Constrained manipulators; Motion/force control; Adaptive control; Disturbance attenuation 1. Introduction From the pioneering work (McClamroch & Wang, 1988), various strategies were proposed to achieve motion and force control for constrained robots, e.g. Carelli and Kelly (1991), You and Chen (1993), Yao, Chan, and Wang (1994). Most theoretical frameworks stem from a reduced-state position/force control method. Considering parametric uncertain robot manipulators, adaptive control schemes were introduced by Yao and Tomizuka (1995) and Yu and Lloyd (1997). Unfortunately, the reduced dynamics based scheme usually has a residual force tracking error proportional to estimated parameter errors. Thus, a high gain force feedback 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Goro Obinata under the direction of Editor Mituhiko Araki. This work was supported by the National Science Council, R.O.C., under Grant NSC89-2213-E-033-045. ∗ Corresponding author. Tel.: +886-3-2654815; fax: +886-3-2654899. E-mail address: lian@dec.ee.cycu.edu.tw (K.-Y. Lian). 1 Present address: Department of Electronic Engineering, Chien-Kuo Technology University, Changhua, Taiwan. 0005-1098/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.06.017 or the persistent excitation condition is needed. As stated in Yuan’s works (Yuan, 1997), the above setback has been removed and both zero motion and force tracking errors are obtained. An alternative approach (Yao & Tomizuka, 1995) uses a nonlinear error space decomposition to obtain the same results. In the presence of unstructured uncertainties of the robot arm, several robust control strategies (Hwang & Hu, 2000; Kwan, 1996; Su, Stepanenko, & Leung, 1995; Wang, Soh, & Chean, 1995; Yao et al., 1994; Zhen & Goldenberg, 1996) provide asymptotic motion tracking and an ultimate bounded force error. In addition, Kiguchi and Fukuda (1999) and Song and Cai (1998) accordingly utilize the fuzzy neural networks and variable structure control to compensate the environmental friction. We especially note here that Yoshikawa and Sudou (1993) developed a dynamic position/force control scheme to deal with an unknown constraint. The difficulty for considering an uncertain constraint is that the motion and force control designs are no longer decoupled. In this case, the abovedeveloped adaptation law cannot be applied. This led to complex strategies in Hu, Ang, and Krishnan (2000), and Pagilla and Yu (2001). 2112 C.-S. Chiu et al. / Automatica 40 (2004) 2111 – 2119 Most robust approaches, e.g. Song and Cai (1998), Zhen and Goldenberg (1996), etc., were proposed by utilizing discontinuous control laws, which arise some drawbacks. To the best of our knowledge, not many works provide a disturbance attenuation for both motion and force tracking based on continuous control laws. For example, Chang and Chen (2000) applied the adaptive fuzzy technique to achieve robust motion tracking performance for constrained mechanism. Although vast researches of the H ∞ control technique (Chen, Lee, & Feng, 1994; Chen, Chang, & Lee, 1997; Lee & Cheng, 1996; Tomei, 2000) have be conducted, all the methods did not deal with a robust performance criterion involving non-dynamical variables like force errors. Consequently, the disturbance attenuation problem for force error is not trivial. Considering uncertainty in both environment and robot dynamics, this paper presents an adaptive scheme to obtain robust motion/force tracking. Since the motion/force decomposition control is not valid with constraint surface uncertainties, we develop a nonlinear force filter to assure the boundedness of force error and filtered force error. Since the residual force error depends on the estimated parameter error, the magnitude of the parameter error is concerned in our design. For both disturbance and parameter error attenuations, the robust performance criterion introduced by Tomei (2000) is modified here. Next, combining a typical adaptive controller and an auxiliary control law provides a robust tracking performance for the closed-loop system. After solving a simple linear matrix inequality for gain conditions, arbitrary disturbance attenuation to both motion and force error is obtained. In addition, if an augmented disturbance to the closed-loop system and the estimated parameter error are square-integrable, then a strongly asymptotic motion tracking is achieved while the force error is proportional to the inversion of control gain. The remainder of this brief paper is organized as follows. The dynamics and related properties for constrained robot manipulators are addressed in Section 2. In Section 3, a robust motion/force tracking problem is formulated, where an adaptive controller is developed. Section 4 shows the simulation results for controlling a planar revolute robot in constrained motion. Finally, some conclusions are made in Section 5. 2. Dynamics formulation Consider an n-rigid-link robot working in a constrained motion. Let q ∈ R n denote the joint coordinate of the robot arm. When the robot end-effector is rigidly in contact with an uncertain surface, the environmental constraint is expressed as an algebraic equation of the coordinate q and time t, namely,
(q, t) = 0, where
: R n × R → R m . Denote A(q, t) as the Jacobian matrix of
(q, t) with respect j
(q,t) to q, i.e., A = jq . Without loss of generality, the uncertain constraint is decomposed into a nominal part
∗ (q) and a constraint modeling error part 
(q, t). This implies that the Jacobian matrix A(q, t) = A∗ (q) +
A(q, t) with nominal A∗ (q) and uncertain
A(q, t). Meanwhile, several assumptions on the system are made as follows. Assumption 1. The Jacobian matrix A is bounded for all q and t. Assumption 2. The robot manipulator is operated away from any singularity. In this case, the Jacobian A is of full row rank m, such that the joint coordinate q is partitioned into q = [q1⊤ q2⊤ ]⊤ for q1 ∈ R n−m and q2 ∈ R m , where q2 = (q1 , t) with a nonlinear mapping function (·) from an open set  ⊂ R n−m × R to R m . j j2  j j2  Assumption 3. The terms jq1 , jq 2 , jt , and jt 2 are 1 bounded in the work space. Remark 1. The deformation at the contact and the dynamic interaction between the robot and the environmental constraint are important issues in dealing with force control problem and are usually coped with by impedance control approaches, cf. Doulgeri and Arimoto (1999), and Jung and Hsia (2000). Based on our setting, the modeling error part 
(q, t) can accommodate the uncertainty due to the deformation and the dynamic interaction once Assumptions 1–3 are satisfied. Since the dimension of the constraint is m, the configuration space of the robot is left with n–m degrees of freedom. Based on the full row rank for A, the existence of (q1 , t) in Assumption 2 is assured by the implicit function theorem (cf. You & Chen, 1993; Yuan, 1997). Consider (q1 , t) consist of a nominal part ∗ (q1 ) and an uncertain part (q1 , t), the position and velocity kinematic transformation are given as
 q1 q= , (1) ∗ (q1 ) + (q1 , t) q̇ = J q̇1 + v, (2) ⊤ 1 ,t) ⊤ where J = [ In−m j j(q ] and v = q1 ⊤ j (q ,t) ⊤ 1 ⊤ ] . From the above equations and the [ 0(n−m)×1 jt j
(q ,t) 1 ˙ (q, q̇, t)=A(q, t)q̇ + constraint of velocity,
=0, it jt j
(q ,t) 1 = 0. implies that A(q, t)J (q, t) = 0 and A(q, t)v + jt The dynamic equation of the constrained robot is written as M(q)q̈ + C(q, q̇)q̇ + F (q, q̇, t) =  + A⊤ , (3) where M(q), C(q, q̇)q̇,  are the inertia matrix, Coriolis/centripetal force, and joint driving torque, respectively; F (q, q̇, t) denotes a combinational term of gravitational force, joint stick-slip friction force, and external joint/environment disturbances; and  ∈ R m is a generalized force multiplier associated with the constraint and presents a reaction force in normal direction of the surface. For the controller design, let [E1 | E2 ] be a partition of 2113 C.-S. Chiu et al. / Automatica 40 (2004) 2111 – 2119 an identity matrix In with E1 = [ In−m 0 ]⊤ ∈ R n×(n−m) and E2 = [0 Im ]⊤ ∈ R n×m . Define an invertible matrix L = [J E2 ] ∈ R n×n . The velocity transformation (2) then becomes q̇ = LE 1 q̇1 + v. The dynamic equation (3) will be represented in terms of independent coordinate q1 as M(q1 , t)LE 1 q̈1 + (M L̇ + C(q1 , q̇1 , t)L)E1 q̇1 + M v̇ + Cv + F (q1 , q̇1 , t) =  + A⊤ , where the kinematics (1) is used. Multiplying L⊤ on both sides of the above equation leads to M(q1 , t)E1 q̈1 + C(q1 , q̇1 , t)E1 q̇1 + F (q1 , q̇1 , t) = L⊤  + L⊤ A⊤ , (4) where M = L⊤ ML; C = L⊤ (M L̇ + CL); and F = L⊤ (F + M v̇ +Cv). According to that LE 1 =J and AJ =0, a reduced dynamics is obtained after multiplying E1⊤ on both sides of (4): Property 4. The nominal plant has a linear parameterization (LP) form: M∗ (q1 )u̇ + C∗ (q1 , q̇1 )u + F∗ (q1 , q̇1 ) = Y (q1 , q̇1 , u, u̇), where u is an intermediate variable; Y (·) is an n × r known regression matrix; and the vector  ∈ R r consists of unknown parameters of the robot (cf. de Wit, Siciliano, & Bastin, 1997). In addition, each element of  has an upper bound Mi , for i = 1, 2, ..., r. Property 5. Under Assumption 3 along with a bounded disturbance, the terms
J ,
J˙,
M,
C, and
F have upper bounding functions: 
M + J ⊤ M∗
J 
bm (q1 , t), 
C + J ⊤ (M∗
J˙+C∗
J )
bc1 (q1 , t)+bc2 (q1 , t)q̇1 , 
F 
bf 1 (q1 , t) + bf 2 (q1 , t)q̇1 , where bm , bcj , bfj , with j = 1, 2 are positive scalar functions bounded in work space. M(q1 , t)q̈1 + C(q1 , q̇1 , t)q̇1 + F (q1 , q̇1 , t) = J ⊤ , with M = E1⊤ ME1 ; C = E1⊤ CE1 ; and F = E1⊤ F . We assume that the system matrices can be decomposed into a linear-in-parameter part and a nonparametric part, i.e., M(q1 , t)=M∗ (q1 )+
M(q1 , t), C(q1 , q̇1 , t)=C∗ (q1 , q̇1 )+
C(q1 , q̇1 , t), and F (q1 , q̇, t) = F∗ (q1 , q̇1 ) +
F (q1 , q̇1 , t), where each element of M∗ , C∗ , F∗ is linear in system parameters and
M,
C,
F are modeling error parts. The feasibility of the decomposition is related to the non-exact coordinate transformation and the intrinsic unstructured uncertainties. Note here that
F may present the external disturbances and the nonparametric friction term. Meanwhile, J = J∗ +
J with a nominal part J∗ and an uncertain part
J defined by J∗ =
In−m j∗ (q1 ) jq1  ,
J = E 2
J 2 , j(q ,t) with
J2 = jq11 being upper bounded by 
J2 
 for 0
 < ∞. Following the above discussion, matrices M, C, F are further represented as M = M ∗ +
M, C = C ∗ +
C, F = F ∗ +
F , where M ∗ = J ⊤ M∗ J ,
M = J ⊤
MJ , C ∗ = J ⊤ (M∗ J˙ + C∗ J ),
C = J ⊤ (
M J˙ +
CJ ), F ∗ = J ⊤ F∗ ,
F = J ⊤ (
F + M v̇ + Cv). Some useful properties for the system are addressed below. Property 1. M(q1 , t) is a symmetric positive-definite matrix. Property 2. M(q2 , t) and M(q2 , t)−1 are uniformly bounded. ˙ − 2C is a skew-symmetric matrix, i.e., Property 3. M ˙ p⊤ (M − 2C)p = 0, ∀p ∈ R n . 3. Robust adaptive motion/force tracking controller For a constrained robot, the control objective is to let the robot end-effector move along a desired trajectory while maintaining a desired normal force on the contact between the end-effector and surface. Here the preplanning motion trajectory and desired normal force are accordingly denoted as q1d (t) and d (t). For controller synthesis, we define some notations: em = q1d − q1 , em ∈ R n−m ,   = d − ,   ∈ Rm, qa = m em + q̇1d , qa ∈ R n−m , where em , ˜ , qa are the motion error, force error, and auxiliary signal vector, respectively; and m is a symmetric positive-definite matrix. Then a motion error measure is given as s =qa − q̇1 = m em + ėm . The set satisfying s(t)=0 will denote an exponentially stable manifold, so that em and ėm converge to zero as long as limt→∞ s(t)=0. Furthermore, a nonlinear force filter presents the effect of force error in the form ėf + (1 +  2 ) f ef = ˜ , (5) with a symmetric positive-definite matrix f and a positive constant . Then the design scheme is to achieve both zero motion/force error as long as s = 0 and ef = 0. However, when considering an uncertain robot system, the above approach is not trivial. An alternative scheme is to attenuate the effect of uncertainties and disturbances to a prescribed level, i.e., the L2 -gain from modeling error to motion/force tracking errors must be less than a prescribed level. Thus, we consider an adaptive tracking performance 2114 C.-S. Chiu et al. / Automatica 40 (2004) 2111 – 2119 criterion consisting of both motion/force errors and formulate the tracking problem as follows: Robust adaptive tracking problem: Given any bounded desired force trajectory and any smooth bounded desired motion trajectory, an uncertain robot system (3) uses a control law  = (t, s, ef ,  , K, kf ),  ˙ = (q1 , q̇1 , qa , q̇a , s),  is an estimated parameter and K, kf > 0 are control where  gains, such that (i) tracking errors containing s, ef ,   and   error of estimated parameter (t) =  − (t) are bounded, when an equivalent unstructured disturbance to the closedloop system d(t) is bounded; (ii) arbitrary disturbance attenuation is achieved in the following performance criterion:  T 2 e⊤ Qe dt t0
1 (t0 ) +  T t0  2 1  k (t)  + d(t)2 dt, (6) ⊤ e⊤  for T > t0 , where e = [s ⊤ em ef⊤ ]⊤ ; 1/ > 0 is a f prescribed level of disturbance attenuation, Q is a positive semi-definite matrix, 1 is a nonnegative constant depending on initial values of s, em , ef ,   and k is a positive con(t) are L2 integrable, then em , ėm , ef stant; (iii) if d(t) and  asymptotically converge to zero while force tracking error is as small as the inversion of control gain kf . Now the above-tracking problem is to be solved. Based on the defined error measure, an error dynamics in terms of s is written as ME1 ṡ = ME1 q̇a − ME1 q̈1 = − CE1 s+ME1 q̇a +CE1 qa +F −L⊤ (A⊤ +) = −CE1 s+L⊤ (Y (q1 , q̇1 , J∗ (q1 )qa , dtd (J∗ (q1 )qa )) ⊤ ⊤ − A  +
A d − ) + w, (7) where Property 4 has been used in Y (·) = M∗ (q1 ) dtd (J∗ (q1 )qa ) + C∗ (q1 , q̇1 )J∗ (q1 )qa + F∗ (q1 , q̇1 ), and w = ME1 q̇a + CE1 qa + F − L⊤ (Y  +
A⊤ d ), which is bounded above through the following inequality (for proof see Appendix A): s ⊤ E1⊤ w
d12 (q1 , t) + d2 (q1 , t)s2 + d3 (q1 , t)e2 + d4 (q1 , t)s2 e2 , (8) where dj , j = 1, 2, 3, 4, are positive functions depending on the desired trajectories (i.e., q1d , q̇1d , q̈1d ), control gain m , and the structure of the robot. Then, the robust adaptive control law is set in the form:  = Y  − A⊤ ∗ (d + kf ef ) + E1 (Ks + v ), (9) where kf > 0, K ∈ R (n−m)×(n−m) is symmetric and positive-definite, and v is an auxiliary control law determined later. The estimated parameter vector is tuned by  ˙ = Proj( , ), for all | i (t0 )| < Mi , i = 1, 2, ..., r, (10) where  = Y ⊤ J∗ s,  = ⊤ > 0, and the projection operator is defined as  i if ℓ( i )
0,  ˙ =   (11)  if ℓ( i )  0and i i
0, i i  (1 − ℓ( i ))i otherwise, 2 with | i (t0 )| < Mi , ℓ( i ) = ( i − 2Mi )/(ε2 + 2εMi ) and a given positive constant ε. Note that the projection algorithm introduced by Pomet and Praly (1992) has the properties: (a) | i (t)| < Mi +ε for all t  t0 , (b) Proj( , ) is Lipschitz and ⊤ ⊤    continuous, (c)  Proj(, )    for   =  − . Therefore, the error dynamics (7)becomes ME1 ṡ = − CE1 s − E1 (Ks + v )+L⊤ (Y +A⊤ ( +kf ef ) − kf
A⊤ ef )+w, (12) where the fact L⊤ E1 = E1 is used. For a constrained robot system (3) perturbed by bounded disturbances, we give the bounded desired motion trajectories q1d , q̇1d , q̈1d and force trajectory d . When the controller (9) with update law (10) is applied, the stability and auxiliary control law design are given below. , and   are bounded Lemma 1. Errors em , ėm , ëm , ef , ėf ,  if (i) the auxiliary control law v is set as v = 2Pm em + 41 k 2 Y2 2 s + km em 2 s, (13) where Y2 =Y ⊤ E2 , k > 0, km  supq1 , t d4 (q1 , t), and matrix Pm is symmetric positive-definite; and (ii) an appropriate choice of K, m , f , kf satisfies min (K) > supq1 ,t d2 (q1 , t) + kf 
AJ 2 (14) ≡ k, ⊤ min ( m Pm + Pm m ) > supq1 ,t d3 (q1 , t) ≡ d3 , (15) min ( f)> max 1 4 kf , (16) , where the operator min (·) denotes the minimum eigenvalue of a matrix. Proof. Consider the Lyapunov function candidate ⊤ ⊤ Pm em + 21 ef⊤ ef + 21  −1 . V = 21 s ⊤ E1⊤ ME1 s + em Using the error dynamics (12), Property 3, and property (c) of update law (10) renders the time derivative of V to ˙ − 2C)E s − s ⊤ Ks V̇ = 21 s ⊤ E1⊤ (M 1 ⊤ ⊤ −1 − s  +   (Y J s −  ˙ ) + s ⊤
J ⊤ Y ⊤  ∗ v 2 2 + s ⊤ J ⊤ A⊤ (  + kf ef ) − kf s ⊤ J ⊤
A⊤ ef + s ⊤ E1⊤ w ⊤ + 2em Pm ėm + ef⊤ ėf
− s ⊤ Ks − s ⊤ (v − 2Pm em − 41 k
J2⊤ Y2⊤ Y2
J2 s) 2 − kf s ⊤ J ⊤
A⊤ ef + s ⊤ E ⊤ w + 1  k ⊤ − em ( ⊤ m Pm 2 1 + Pm − (1 +  )ef⊤ m )em ⊤ f e f + ef  . (17) 2115 C.-S. Chiu et al. / Automatica 40 (2004) 2111 – 2119 As shown in Appendix, the term s ⊤ E1⊤ w is bounded by (8). This results in V̇ along with v in (13) as V̇
− (min (K) − d2 − kf 
AJ 2 )s2 + Pm m ) − d3 )em 2 2 1 f ) − 4 kf )ef  2 + d2 , (18) − (min ( f ) − ) 2 ef 2 + k1  − (min ( − (min ( ⊤ m Pm  d2 = d12 1 4 + where is an equivalent disturbance of closedloop system. If the gain conditions in (14)–(16) are satisfied, there exist 1 , 2 , 3 > 0 such that (18) is further bounded by V̇
− 1 s2 − 2 em 2 − 3 ef 2 + 1  2 k  ṡ = M (−Cs − Ks − v + J Y  ⊤ ⊤ ⊤ − kf J
A ef + E1 w). ⊤ (20) All terms on the right-hand side of (20) are bounded, which implies ṡ ∈ L∞ and ëm ∈ L∞ . According to (12), (20), and a full row rank matrix A, the force tracking error is expressed as −1 ⊤  ⊤   = − kf ef +A−⊤ 2 E2 (ME1 M (−Cs−Ks−v +J Y  − kf J ⊤
A⊤ ef + E1⊤ w)  + kf
A⊤ ef − w) + CE1 s − Y ≡ − kf ef + z(q1 , q̇1 , s, em , ėm , ef ,  , t), (21) ⊤ ⊤ ⊤ ⊤ ⊤ where A⊤ 2 = E2 A and E2 L = E2 have been used. Con sequently,  is bounded based on the above results. This also implies ėf is bounded from the force filter (5).  Lemma 2. The disturbance attenuation of (6) is achieved after the following linear matrix inequality problem is solved: Given m > 0, , , kf > 0, Q  0, there exist K > 0, Pm > 0, f > 0 such that P− 2 Q0 (22) with P = block-diag{K − kIn−m , d 3 In−m , f − 41 kf Im , f − Im }. ⊤ m Pm + Pm m − Proof. Based on the results of Lemma 1, (17) is written as V̇
− e⊤ P e + 1  2 k  + d2 , V̇
− 2 ⊤ If P − 2 Q  0, V̇
− 2 ⊤ e Qe − e⊤ (P − 2 Q)e + 1  2 k  + d2 . the time derivative of V further yields 1  2 k  e Qe + + d2 . (23) Integrating both sides of inequality (23), we obtain 2 + d2 . (19) Since the projection in (10) guarantees the boundedness of  , we conclude that   ∈ L∞ . Thus, if the disturbance is bounded, then V̇ is negative semidefinite in the region S =  2 − {s, em , ef  | 1 s2 + 2 em 2 + 3 ef 2 − k1   d2 > 0}. From the positive-definite V and V̇ in (19), we have s, em , ėm , ef ∈ L∞ . In addition, multiplying both sides of error dynamics (12) by E1⊤ follows that −1 ⊤ e⊤  ef⊤ ]⊤ . Add and subtract a term with e = [s ⊤ em f 2 e⊤ Qe to the right-hand side of the above inequality, we have  T e⊤ Qe dt t0
V (t0 ) +  T t0  2 1  k (t)  + d(t)2 dt. (24) Therefore, (24) implies a robust performance in form (6). Note that since the projection operator (10) renders  i (t)
Mi + ε, a conservative result is also given as 2  T t0 e⊤ Qe dt
V (t0 ) + +  T t0 1 k sup  (t)2 (T − t0 ) t∈[t0 ,T ] d(t)2 dt.  Lemma 3. The force error is attenuated to −1 level of z(q1 , q̇1 , s, em , ėm , ef ,  , t) with  = (min ( f ) + kf )/max ( f ). In addition, if d(t) and  (t) are L2 integrable, then em , ėm , ef asymptotically converge to zero while the force tracking error is with a bound proportional to −1 level of  supq1 ,t (q1 , t), for some  > 0 and  = limem ,ėm ,→0 w. Proof. According to the nonlinear force filter (5) and the representation of force error in (21), it follows that   = ( kf (D + (1 +  2 ) −1 f) + Im )−1 × z(q1 , q̇1 , s, em , ėm , ef ,  , t), where (D + (1 +  2 ) f )−1 denotes the nonlinear filter operator. Omitting some higher order terms, the ultimate bounded error of   is approximately determined by −1 z(q1 , q̇1 , s, em , ėm , ef ,  , t), (25) with  = (min ( f ) + kf )/max ( f ). Furthermore, if  (t)  is L2 integrable, then (t) converges to zero as t → ∞ by Barbalat’s lemma, since  ,  ˙ ∈ L∞ and   ∈ L2 have been shown. Thus, both d(t) and  (t) having finite L2 -norm implies that the error vector e belongs to L2 from (23). Combining the results s, ṡ ∈ L∞ , ef , ėf ∈ L∞ , and s, ef ∈ L2 , the motion errors s, em , ėm and filtered force error ef asymptotically converge to zero by Barbalat’s lemma. Therefore, applying the above convergence on the term z(·) in (25), 2116 C.-S. Chiu et al. / Automatica 40 (2004) 2111 – 2119 the approximate force tracking error ⊤ −1 ⊤  ≈ −1 A−⊤  2 E2 (ME1 M E1 w − w) ≈ −1  sup (q1 , t), q1 ,t where  = limem ,ėm ,→0 w and  > 0.  Summarizing the results in Lemmas 1–3, we have the following main theorem. Theorem. Using the control law (9) with the update law in (10) and v in (13), if the control gains are appropriately chosen to satisfy the linear matrix inequality (22), then the robust adaptive motion/force tracking problem is solved for the constrained robot system (3).  Accordingly, the design procedure first begins at the force tracking error attenuation by choosing and kf of (25). Then the effect from the equivalent disturbance to the motion errors and filtered force error are attenuated to 1/ level based on Lemma 2. Since the gain conditions arise from the conservative stability analysis procedure, the sufficient condition of (14)–(16) can be used as a guideline rather than an absolute mandate. Remark 2. When considering a limited computational capacity in the implementation, the projection (11) can be simplified to i if  i (t)
Mi ,  ˙ i (t) = 0 otherwise. This yields a small update time interval but is with the same robust result as the main theorem. Furthermore, instead of on-line calculating the norm of the matrix Y2 in the auxiliary control law v , a pre-found upper bound of Y2 would be used once the control system works in a local space of q.  4. Simulation results A two-link revolute robot manipulator in constrained motion is considered for numerical simulations, as shown in Fig. 1. The end-effector moves along an uncertain straight line on a vertical plane, i.e., the constraint surface in the Cartesian coordinates is x = 1 + 0.01 sin(3t)m and y  0. When the length of the first link has deviation l1 = 1 ± 0.05 m and the length of the second link is exactly known as l2 = 1 m, the forward kinematics and its Jacobian are 

x l cos q1 + cos(q1 + q2 ) , = 1 y l1 sin q1 + sin(q1 + q2 )
 −l1 sin q1 − sin(q1 + q2 ) − sin(q1 + q2 ) Jq = , l1 cos q1 + cos(q1 + q2 ) cos(q1 + q2 ) which implies that the constraint equation, i.e.,
∗ (q) + 
(q, t) = 0, has
∗ (q) = cos(q1 ) + cos(q1 + q2 ) − 1, 
(q, t) = ± 0.05 cos(q1 ) − 0.01 sin(3t). Then A(q, t) is composed of A∗ = [− sin(q1 ) − sin(q1 + q2 ) − sin(q1 + q2 )] and
A = [±0.05 sin(q1 ) 0]. The position transformation (q1 ) consists of ∗ (q1 ) = cos−1 (1 − cos(q1 )) − q1 , (q1 , t) = cos−1 (1 + 0.01 sin(3t) − l1 cos(q1 )) − cos−1 (1 − cos(q1 )), for 0
1+0.01 sin(3t)−l1 cos(q1 )
1. The velocity transformation (2) is thus defined with 
1 sin(q ) , J∗ = −1 − √ 1 1−(1−cos(q1 ))2  −(1±0.05) sin(q1 )
J = E 2 √ 1−(1+0.01 sin(3t)−(1±0.05) cos(q1 ))2  , + √ sin(q1 ) 2 v 1−(1−cos(q1 )) −0.03 cos(3t) . = E2 √ 1−(1+0.01 sin(3t)−(1±0.05) cos(q1 ))2 In addition, system (3) has the following system matrices: 
 + 22 cos q2 3 + 2 cos q2 M∗ (q) = 45 1 , 3 + 2 cos q2 3
M = 41 M∗ (q),
4 −2 q̇2 sin q2 C∗ (q, q̇) = 5 2 q̇1 sin q2  −2 (q̇1 + q̇2 ) sin q2 , 0
C(q, q̇) = 41 C∗ (q, q̇)
  g cos q1 + 3 g cos(q1 + q2 ) F∗ (q, q̇) = 45 4 3 g cos(q1 + q2 )
 5 q̇1 + 6 sign(q̇1 ) + , 5 q̇2 + 6 sign(q̇2 )
 1 4 g cos q1 + 3 g cos(q1 + q2 )
F∗ (q, q̇, t) = 5 + d , 3 g cos(q1 + q2 ) where g = 9.81, 1 = (m1 + m2 )l12 + m2 , 2 = m2 l1 , 3 = m2 , 4 = (m1 + m2 )l1 (l2 = 1 is used), 5 , 6 are friction parameters, and d is an external disturbance defined below:
 ϑ1 sign(q̇1 ) exp(−|q̇1 |/ϑ2 ) d = ϑ1 sign(q̇2 ) exp(−|q̇2 |/ϑ2 )
 (ϑ3 sign(ẏ) + ϑ4 ẏ) + d0 . + Jq⊤ 0 Note that the second term of F∗ (·) and the first term of d denote the joint friction force, where the nonparametric Stribeck effect of friction is considered in d . The second term of d is a friction force in tangent direction of the surface. In these terms, ϑ1 ∼ ϑ4 are friction parameters. The term d0 , set as d0 = 0.2 sin(200t)[1 1]⊤ , exhibits the effect of high frequency measurement noise. Here we set the 2117 C.-S. Chiu et al. / Automatica 40 (2004) 2111 – 2119 3 2.5 environment surface (x, y) q⋅ 1 (t) 2 m2 1.5 rad/sec l2 q2 q⋅ 1d (t) 1 0.5 0 m1 −0.5 l1 −1 q1 −1.5 x=1 0 2 4 6 8 10 t (sec) Fig. 1. A two-link planer constrained robot manipulator. Fig. 3. Response of joint velocity of the first joint. 3 q1d (t) 1 2.5 2 1.5 ~ λ (t) 1 0.6 N rad 0.8 0.5 0 0.4 −0.5 q1 (t) 0.2 −1 −1.5 0 0 2 4 6 8 10 t (sec) −2 0 4 6 8 10 t (sec) Fig. 2. Response of joint position of the first joint. actual values of 1 ∼ 6 , ϑ1 ∼ ϑ4 as 1 =3.7075, 2 =0.95, 3 = 1, 4 = 2.85, 5 = 0.2, 6 = 0.5, ϑ1 = 0.1, ϑ2 = 0.1, ϑ3 =0.2, and ϑ4 =0.1, respectively. Let the parametric vector be  = [1 2 3 4 5 6 ]⊤ . Each term i has a known upper bound. This implies that the upper bound of d1 ∼ d4 can be found. To perform the robust adaptive controller, the parameter ε of the projection of update law (10) is 0.5. The initial states are set as q1 (0) = 0.524, q2 (0) = 0.913, and q̇1 (0) = q̇2 (0) = 0. According to the nominal constraint, the desired motion  trajectory q1d = 4 + 12 sin(t) while the desired normal force is d = 5 + sin(t). The control gains are chosen as = 5, kf = 20, k = 5, km = 5, m = 10, = 2, Q = I4 , and  = 0.5I6 . Thus, (22) is solved and f = 10, K = 20, and Pm = 5 are obtained. The simulation results for the motion tracking of joint position and velocity are shown in Figs. 2 and 3, respectively. The desired force and constraint force are shown in Fig. 4. Notice that the abrupt impulses arise mainly due to the disturbance of friction force. Additionally, 2 Fig. 4. Force tracking error. all estimated parameters are shown in Fig. 5. The control inputs of two joints of the robot are illustrated in Fig. 6. 5. Conclusions An adaptive tracking controller has been designed for constrained robots subject to nonexactly known constraint, disturbances, and modeling uncertainties, where all the position, velocity, force, and filtered force errors are guaranteed to satisfy a robust tracking performance criterion. Although the motion/force decomposition is no longer valid, the proposed nonlinear force filter provides the boundedness of motion and force errors in stability analysis. Using the joint position, joint velocity, and normal contact force as the available measurements, the proposed controller consists of an adaptive controller part and an auxiliary control law part with some nonlinear damping terms. The disturbance and parameter error attenuation is guaranteed in a 2118 C.-S. Chiu et al. / Automatica 40 (2004) 2111 – 2119 robot. First, we rewrite E1⊤ w as 3.5 E1⊤ w = J ⊤ (Y (q1 , q̇1 , J (q1 , t)qa , dtd (J (q1 , t)qa )) θ^4 (t) θ^1 (t) 3 − Y (q1 , q̇1 , J∗ (q1 )qa , dtd (J∗ (q1 )qa ))) 2.5 = (
M + J ⊤ M∗
J )q̇a + (
C + J ⊤ (M∗
J˙ + C∗
J ))qa +
F − J ⊤
A⊤ d . 2 Then Property 5 renders the norm of E1⊤ w to 1.5 1 θ^ 2 (t) 0.5 0 θ^3 (t) E1⊤ w
bm (m ėm  + d2 ) + (bc1 + bc2 q̇1 )(m em  + d1 ) θ^6 (t) + bf 1 + bf 2 q̇1  + J ⊤
A⊤ d  θ^5 (t) 0 2 4 6 8 10 t (sec) Fig. 5. Response of estimated parameters. with d1 = supt q̇1d ∞ , d2 = supt q̈1d ∞ , and m = min ( m ). Furthermore, applying the facts, q̇1  = q̇1d − ėm 
d1 + ėm  and ėm  =  m em − s
m em  + s, yields E1⊤ w
bm (2m em  + m s + d2 ) + (bc1 + bc2 (d1 + m em  + s))(m em  + d1 ) + bf 1 150 + bf 2 (d1 + m em  + s) + J ⊤
A⊤ d  Nm 100 which can be regrouped as E1⊤ w 50
(bm d2 + bc1 d1 + bc2 2d1 + bf 1 + bf 2 d1 + J ⊤
A⊤ d ) + (bm m + bc2 d1 + bf 2 )s + bc2 m em s + (bm 2m + bc1 m + 2bc2 d1 m 0 −50 + bf 2 m )em  + bc2 2m em 2 0 2 4 6 8 10 t (sec) Fig. 6. Control inputs in the first joint (solid) and second jont (dashed). ≡ b1 + b2 s + b3 sem  + b4 em  + b5 em 2 . Thus, using the upper bound of E1⊤ w results in |s ⊤ E1⊤ w|
b1 s + b2 s2 + b3 s2 em  global fashion by solving a linear matrix inequality problem, i.e., the effect from disturbance and estimated parameter errors to motion/force errors is attenuated to an arbitrary prescribed level. Furthermore, under the condition of finite square-integral for disturbance and estimated parameter error, the motion tracking error is concluded in an asymptotic sense and the force tracking error is as small as the inversion of control gain. Finally, taking a two-link constrained planar robot as an example, the simulation results have shown the expected performance. 6. Appendix. Boundedness of s ⊤ E1⊤ w In this appendix, we will show that the term s ⊤ E1⊤ w in (17) has an upper bounded function which depends on the position error em , error measure s, and the structure of the + b4 sem  + b5 sem 2 . Recursively applying the triangle inequality on the above equation, we have   |s ⊤ E1⊤ w|
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Combined adaptive and variable structure control for constrained robots. Automatica, 31(3), 483–488. Tomei, P. (2000). Robust adaptive friction compensation for tracking control of robot manipulators. IEEE Transactions on Automatic Control, 45(6), 2164–2169. Wang, D., Soh, Y. C., & Chean, C. C. (1995). Robust motion and force control of constrained manipulators by learning. Automatica, 31(2), 257–262. Yao, B., Chan, S. P., & Wang, D. (1994). Variable structure adaptive motion and force control of robot manipulators. Automatica, 30(9), 1473–1477. 2119 Yao, B., & Tomizuka, M. (1995). Adaptive control of robot manipulators in constrained motion controller design. ASME Journal of Dynamics System Measures & Control, 117, 320–327. Yoshikawa, T., & Sudou, A. (1993). Dynamic hybrid position/force control of robot manipulators—on-line estimation of unknown constraint. IEEE Transactions on Robotics and Automation, 9(2), 220–226. You, L.-S., & Chen, B. -S. (1993). Tracking control design for both holonomic and non-holonomic constrained mechanical systems: a unified viewpoint. International Journal of Control, 58(3), 587–612. Yu, H., & Lloyd, S. (1997). Combined direct and indirect adaptive control of constrained robots. International Journal of Control, 68(5), 955–970. Yuan, J. (1997). Adaptive control of a constrained robot—ensuring zero tracking and zero force errors. IEEE Transactions on Automatic Control, 42(12), 1709–1714. Zhen, R. Y., & Goldenberg, A. A. (1996). Variable structure hybrid control of manipulators in unconstrained and constrained motion. ASME Journal of Dynamic of System, Measurement and Control, 118, 327–332. Chian-Song Chiu received the B.S. degree in electrical engineering and the Ph.D. degree in electronic engineering both from the Chung-Yuan Christian University in 1997 and 2001, respectively. He is now an Assistant Professor at Chien-Kuo Institute of Technology, Changhua, Taiwan, ROC, since 2003. His current research interests are in robotics, fuzzy systems, and nonlinear control. Kuang-Yow Lian received the B.S. in engineering science in 1984 from National Cheng-Kung University and the Ph.D. degree in 1993 in electrical engineering from National Taiwan University, Taiwan. From 1986 to 1988, he served as a control engineer at ITRI. He joined Chung-Yuan Christian University, Taiwan in 1994. He is currently a professor and chair for the Department of Electrical Engineering. He is the recipient of Outstanding Research Award of the university. His research interests include nonlinear control systems, fuzzy control, robotics, chaotic systems, and nonholonomic control. Tsu-Cheng Wu received the B.S. degree in applied mathematics in 1994 from National Chiao-Tung University and M.S. degree in the electrical engineering in 2002 from Chung-Yuan Christian University. He is currently studying for a Ph.D. degree at Chung-Yuan Christian University. His research interests include network control and nonlinear control systems.