Adaptive Dynamic Surface Control of a Exible-Joint Robot with Parametric Uncertainties

Scientia Iranica B (2019) 26(5), 2749{2759

Sharif University of Technology Scientia Iranica Transactions B: Mechanical Engineering http://scientiairanica.sharif.edu

Adaptive dynamic surface control of a exible-joint robot with parametric uncertainties

C.G. Li, W. Cui, D.D. Yan, Y. Wang, and C.M. Wang

College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China.

Received 8 January 2017; received in revised form 12 February 2018; accepted 14 May 2018

KEYWORDS Abstract. A new kind of adaptive Dynamic Surface Control (DSC) method is proposed in this study to overcome parametric uncertainties of Flexible-Joint (FJ) robots. These FJ robot; uncertainties of FJ robots are transformed into linear expressions of inertial parameters, Dynamic surface which are estimated based on the DSC, and the high-order derivatives in DSC are solved by control; using a rst-order lter. The adaptation laws of inertial parameters are designed directly to Inertial parameters; improve the tracking performance according to the Lyapunov stability analysis. Simulation Adaptive control; results of a two-link FJ robot show better tracking accuracy against model parametric Tracking accuracy. uncertainties. The used method does not need the aid of Neural Network (NN); it is simpler and calculates faster than the other adaptive methods. © 2019 Sharif University of Technology. All rights reserved.

1. Introduction heavy computing burden [4,5]. To overcome this complexity, the DSC is investigated by introducing a rst- Flexible-Joint (FJ) robots equipped with harmonic order lter in the backstepping procedure [6]. Because drivers, windup shaft, and force/torque sensors are of the great approximation capability of nonlinear func- used widely in the areas such as aerospace, service tions, the Neural Networks (NNs) are widely utilized robots, and so on [1-3]. Because of the uncertain in the control systems to compensate the uncertainties dynamic parameters such as inertia, exibility, and of parameter and are combined with the DSC and the external disturbances, the tracking performance of backstepping techniques to design the controller [7- robots is not ideal generally and, even, the systems are 14]. In [14], NN adaptive backstepping controller unstable under basic control methods. was proposed, and Radial Basic Function (RBF) was In the last decades, many researchers have pro- used to approximate the nonlinear unknowns in the posed and developed the backstepping control tech- backstepping design. Simulation results showed good nique because of its advantages such as systematic tracking performance. and recursive design methodology for nonlinear control. Due to the adaptiveness of fuzzy, it was em- However, its disadvantage is \explosion of complexity" ployed too in the control design [15-19]. In [15], or \explosion of derivative" caused by the repeated the backstepping control design was suggested, and di erentiation of virtual control vectors, resulting in the adaptive fuzzy DSC was applied to approximate unknown nonlinear control system, which was more *. Corresponding author. Tel.: 86 25 84892503 general for practical applications in the presence of E-mail addresses: [email protected] (C.G. Li); input saturation. The prescribed performance of [email protected] (W. Cui); [email protected] (D.D. Yan); [email protected] (Y. switched adaptive DSC was investigated for a class Wang); [email protected] (C.M. Wang) of switched nonlinear systems, and mode-dependent fuzzy logic systems were used to approximate the doi: 10.24200/sci.2018.20492 switching nonlinear functions [16]. The adaptive fuzzy 2750 C.G. Li et al./Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2749{2759 dynamic surface control was investigated for a class of and adaptive DSC techniques. Then, the adaptation nonlinear systems with fuzzy dead zone and unmodeled laws for these inertial parameters are derived according dynamics. The Takagi-Sugeno-type fuzzy logic systems to the Lyapunov stability analysis, which reduce the were adopted to approximate the unknown functions computing burden eciently. Finally, simulations of in system [17]. [20] employed the Fuzzy Neural Net- the two-link FJ robot show the e ectiveness of the works (FNNs) to approximate uncertain dynamics adaptive DSC technique. and bounding functions when the active DSC was This paper is organized as follows. In Section 2, developed to suppress regenerative chatter in micro- the basic dynamic models and properties of FJ robot milling. are introduced. In Section 3, the backstepping control Some disturbance oservers (DOB) are employed design method is proposed according to the Lyapunov commonly to design nonlinear controllers because of stability theorem. Simulation results and analysis are their distinct physical meaning [21-23]. In [21], a Gain- discussed in Section 4. Finally, some conclusions are Scheduled Dynamic Surface Control (GSDSC) based given in Section 5. on the NN disturbance observer (NNDOB) was proposed for a class of uncertain underactuated mechan- 2. Dynamic model of FJ robots ical systems, which eciently solves the mismatches and overcomes the problem of \explosion of complex- In general, the dynamic model of n-link FJ robot ity". In [22], a nonlinear disturbance observer was consists of robot dynamics and actuator dynamics, constructed to overcome the unknown environmental which can be expressed in the following form [28]: disturbances; then, a controller was designed based on D(q)q + C(q; _q) _q + G(q) + K(q q ) = 0; (1) the combination of backstepping and DSC techniques. m An observer-based DSC scheme was developed to over- Jq + K(q q) = u; (2) come the problems of model uncertainties, unsteady m m aerodynamics, and actuator saturation in [23]. where q; _q; q 2 Rn denote the position, velocity, and Sliding Mode Control (SMC) is one of the e- acceleration vectors of the links, respectively, D(q) 2 cient control schemes for compensating external distur- Rn denotes the inertia matrix, C(q; _q) 2 Rn is the bances and parametric uncertainties. In [24,25], sliding Coriolis and centripetal matrix, G(q) 2 Rn denotes the mode control and adaptive control were introduced into gravity matrix, K 2 Rn represents a positive de nite the backstepping control technique together with DSC. n diagonal constant exibility matrix, qm; qm 2 R Further, [26] designed an adaptive backstepping denote the position and acceleration vectors for the controller with a dynamic surface method to solve actuator, respectively, J 2 Rn is the actuator inertia the problem of parameters of permanent-magnet syn- matrix, and u 2 Rn is the actual control vector of chronous motor; the particle swarm optimization algo- the actuator torque. In dynamic equations (Eqs. (1) rithm was adopted to adjust and determine the control and (2)), the joint sti ness terms are assumed to be parameters. dominant among all of the system parameters, and the Although the above approximate control schemes joint damping ones are neglected. As introduced in enjoy a number of advantages, they are susceptible literature [9], these equations have a few fundamental to certain problems such as heavy calculation burden, properties to make the control system design simple. long training time, approximating error, and so on, re- These properties are presented below. sulting in the diculty choosing the control parameters and guaranteeing system stability. - (P1): The link inertia matrix, D(q), is positive In [27], the dynamic surface adaptive backstep- de nite, symmetric and bounded: ping control scheme was employed to prevent the uncer- D jjD(q)jj D ; tainties of nonlinear dynamics of aircraft. By changing m M the inertial parameters and positions of aircraft center where Dm and DM denote two di erent positive of gravity, the simulation showed the e ectiveness of constants, respectively. the controller. However, the selected parameters were - (P2): The Coriolis and centripetal matrix, C(q; _q), limited and in exible. and gravity matrix, G(q), are bounded, respectively: In this paper, a new adaptive DSC technique for FJ robots is proposed to overcome the problems jjC(q; _q)jj CM ; jjG(q)jj GM ; of parametric uncertainties, explosion of derivative, and calculation ineciency. First, model uncertainties where Dm and DM denote positive constants. of FJ robots are expressed as linear equations of - (P3): The matrix D_ (q) 2C(q; _q) is skew- its inertial parameters; thus, the model uncertainties symmetric. For any vector x, are transformed to compute the evaluation of inertial T _ parameters. The controller is designed by backstepping x (D(q) 2C(q; _q))x = 0: (3) C.G. Li et al./Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2749{2759 2751

- (P4): The system parameters can be linear in the Thus, the parametric uncertainties regarding the equation expressed as coecients of known functions robot model that include D(q), C(q; _q), G(q), K1, of q; _q and q for a rigid joint robot. Therefore, the K, and J can be expressed and solved by using the left-hand side of Eq. (1) can be expressed as follows: vectors i (i = 1; 2; 3; 4). Here, the adaptive DSC algorithm is proposed D(q)q + C(q; _q) _q + G(q) = Y(q; _q; q)1; (4) according to the backstepping control design [4], and the design procedure includes three phases: where Y(q; _q; q) 2 Rnr denotes the regression r1 1. Designing the virtual control vector, , for actua- matrix, 1 2 R is an r-dimensional vector of pa- 1 rameters, and r is the number of inertia parameters. tor position x1; 2. Designing the virtual control vector, , for actua- Further, Eq. (4) can be rewritten according to the 2 tor velocity x ; literature [29] as follows: 2 3. Designing the actual control vector, u. Details are shown as follows: D(q)a1 + C(q; _q)v1 + G(q) = Y(q; _q; v1; a1)1; (5) where a1 and v1 are the arbitrary n-dimensional Phase 1 vectors. As a preliminary to the control design, x1 = qm Choose a virtual control vector 1 for x1 as follows: and x = _q are de ned as the functions of state 1 2 m = q + K^ u ; (9) space variables, and dynamic systems (1) and (2) are 1 r described as follows: where (^) is the estimated value of (), and the auxiliary control vector, ur, is de ned as follows: D(q)q + C(q; _q) _q + G(q) + Kq = Kx ; (6) 1 ^ ^ ^ ur = D(q)a1 + C(q; _q)v1 + G(q) 2r1; (10) _x1 = x2; (7) where v1 = _qd 1~q, a1 = _v1, r1 = _q v1 = _ nn nn ~q + 1~q, ~q = q qd, 1 2 R and 2 2 R _x = J1u J1K(x q): (8) 2 1 are diagonal positive de nite matrices [16], qd is the desired trajectory of link position. To develop the controller for systems (6)-(8), some Eq. (10) can be rewritten by using (P4) as follows: assumptions [8,22] are proposed: ^ ur = Y(q; _q; v1; a1)1 2r1: (11) Assumption 1. All the state space variables can be To avoid the problem of \explosion of derivative", the achieved by some sensors. estimated value of derivative of 1 is derived by using the rst-order lter in the DSC. That is: Assumption 2. The model parameters of D(q), 1 _ 1 + 1 = 1; 1(0) = 1(0); (12) C(q; _q), G(q), K in Eq. (6) are unknown. However, where denotes a diagonal positive de ne matrix, and there are positive constants Km;KM ;Jm;JM to make 1 1 is the ltering control vector. Km jjKjj KM ;Jm jjJjj JM : De ne:

Assumption 3. The desired trajectories of link po- z1 = x1 1: (13) sition q are uniformly bounded, di erentiable up to d By substituting Eq. (13) into Eq. (6), it can be obtained the second order and satisfy the inequality: as follows: 2 2 2 D(q)q+C(q; _q) _q+G(q)+Kq = K + Kz : (14) jjqdjj + jj _qdjj + jjqdjj Q; 1 1 From Eqs. (9), (12), and (14), the following is yielded: where Q is the positive constant. D(q)q + C(q; _q) _q + G(q) + Kq 3. Adaptive DSC system ^ 1 = Kq + KK ur + Kz1 K 1 _ 1 3.1. Controller design The system inertia parameters in FJ robots are clas- ^ 1 1 = Kq + ur + K(K K )ur si ed into four vectors: 1, 2, 3, and 4, which are composed of the elements in the link dynamic + Kz K _ : (15) parametric matrix, the inverse of joint sti ness matrix, 1 1 1 the joint sti ness matrix, and the actuator inertia To get the estimated value of K^ 1, the de nition is matrix, respectively. given by: 2752 C.G. Li et al./Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2749{2759

^ 1 ^ ^ ^ K ur = diag(ur)2; (16) u = K(x1 q) + J_ 2 4z2; (26)

nn where diag() denotes the diagonal matrix composed where 4 2 R denotes a diagonal positive de nite of all the elements of (). matrix. Then, substituting Eq. (16) and (P4) into Eq. (15) By simply considering K^ and J^, such similar yields: expressions as Eq. (16) are given below: ~ ~ D(q)_r + C(q; _q)r + 2r1 = D(q)a + C(q; _q)v ^ ^ 1 1 1 1 K(x1 q) = diag(x1 q)3;

+ G~ (q) + Kdiag(u )~ + Kz K _ ^ ^ r 2 1 1 1 J_ 2 = diag( _ 2)4: (27) ~ ~ From Eqs. (27) and (26), it follows that: = Y(q; _q; v1; a1)1 + Kdiag(ur)2 + Kz1 u = diag(x q)^ + diag( _ )^ z : (28) K 1 _ 1; (17) 1 3 2 4 4 2 where (~) = () (^). By substituting Eqs. (28) and (27) into Eq. (25), it From Eqs. (9) and (16), the equation can be follows that: described as follows: ^ ^ J_z2 = (K K)(x1 q) + (J J) _ 2 4z2 ^ 1 = q + diag(ur)2: (18) ~ ~ = diag(x1 q)3 + diag( _ 2)4 4z2: (29) Phase 2 3.2. Stability analysis Di erentiating Eq. (13) yields: De ne the error-surface vectors as follows: s = ; i = 1; 2: (30) _z1 = _x1 _ 1 = x2 _ 1: (19) i i i

Design the second virtual control vector 2 as in the By using Eqs. (12) and (21), Eq. (30) can be rewritten following: as follows:

2 = _ 1 3z1; (20) si = i _ i; i = 1; 2: (31)

nn where 3 2 R denotes the diagonal positive de nite After taking the rst-order di erential of Eq. (30) with matrix. respect to time and considering Eqs. (9), (20), and (31), In addition, 2 is obtained using the rst-order the equations can be obtained as follows: lter again as follows: _ 1 _s1 = _ 1 1 = 1 s1 + 1; (32) 2 _ 2 + 2 = 2; 2(0) = 2(0); (21) _ 1 _s2 = _ 2 2 = 2 s2 + 2; (33) where 2 is the diagonal positive de nite vector, and 2 is the ltering control vector. ^ where 1 = d(q1 +diag(ur)2)=dt and 2 = 3 _z1 1 are the continuous bounded vectors. De ne: Choose the global Lyapunov function: z2 = x2 2: (22) V = V1 + V2 + V3; (34) Substituting Eqs. (20) to (22) into Eq. (19) yields the following: where, _z = z _ z : (23) 1 1 1 1 2 2 2 3 1 V = rT D(q)r + zT z + zT Jz ; (35) 1 2 1 1 2 1 1 2 2 2 Phase 3 1 T 1 T V2 = s1 s1 + s2 s2; (36) Di erentiating Eq. (22) and substituting it into Eq. (8) 2 2 yields: 1 ~T 1~ 1 ~T 1~ V3 = 1 + K 2 1 1 2 1 1 2 2 2 _z2 = _x2 _ 2 = J u J K(x1 q) _ 2: (24) 1 T 1 T Multiply both sides of Eq. (24) by J, + ~ 1~ + ~ 1~ ; (37) 2 3 3 3 2 4 4 4 J_z2 = u K(x1 q) J_ 2: (25) and i is the diagonal positive de nite matrix, i = Thus, the actual control vector is designed as follows: 1; 2; 3; 4. C.G. Li et al./Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2749{2759 2753

Theorem. By considering the FJ robot descripted as Moreover, the set of inertia parameters i, the esti- ^ ^ Eqs. (1) and (2) with parametric uncertainties and the mated set i, and the set of estimated errors i satisfy initial condition of the Lyapunov function V (0) p (p the following inequality: denotes the arbitrary positive constant), the adaptive laws can be determined using the control law (Eq. (26)) ~T ^ ~T ~ T 2i i i i i i: (44) as follows: 8 By using Eqs. (42), (43) to (44), the derivative of _ _ >^ = ~ = YT r ^ Lyapunov function can be rewritten as follows: > 1 1 1 1 1 1 1 >_ _ <^ = ~ = diag(u )r ^ 2 2 2 r 1 2 2 2 (38) _ T T _ _ V r1 2r1 z1 3z1 >^ = ~ = diag(x q)z ^ > 3 3 3 1 2 3 3 3 :>^_ ~_ ^ 1 4 = 4 = 4diag( _ 2)z2 444 zT z sT 1s sT 1s + rT Kr 2 4 2 1 1 1 2 2 2 4 1 1 where represents the scalar coecients, > 0, and i i 1 i = 1; 2; 3; 4. + zT Kz + rT Kr + sT Ks + zT z 1 1 4 1 1 1 1 1 1 If i; i; i (i = 1; 2; 3; 4) and i(i = 1; 2) are selected properly, it is easy to guarantee the uniformly 1 1 1 + zT z + zT z + sT s + sT s + T semi-globally boundedness of all closed-loop signals, 4 2 2 1 1 4 2 2 1 1 4 1 1 and the tracking errors will converge to a compact set whose size can be adjusted to be arbitrarily small. 1 1 X4 1 + sT s + T + T 2 2 4 2 2 2 i i i 2 i=1;i6=2 Proof. Di erentiate V1 with respect to time and simplify it based on (P3), Eqs. (17), (23), (29), and 4 X T 1 1 T (31). Thus: ~ ~ + T K ~ K~ : (45) i i i 2 2 2 2 2 2 2 2 _ T ~ T ~ T i=1;i6=2 V1 =r1 Y1 + r1 Kdiag(ur)2 r1 2r1 According to (P1) and Assumption 2, Eq. (45) is T T T T + r1 Kz1 + r1 Ks1 z1 3z1 z2 4z2 derived as follows: T T D(q) 1 + z1 z2 + z1 s2: (39) _ T T V r1 2 KM I r1 z1 (3 KM I DM 2 Di erentiate V2 and consider Eqs. (32) and (33): T J 1 T 1 _ T 1 T 1 T T 2I)z1 z2 4 I z2 s1 ( 1 V2 = s1 1 s1 s2 2 s2 + s1 1 + s2 2: (40) JM 4 Similar to di erentiating V3 as above: 5 1 K I I)s sT 1 I s M 1 2 2 4 2 2 _ ~T 1~_ ~T 1~_ ~T 1~_ ~T 1~_ V3 =1 1 1 +2 K2 2 +3 3 3 +4 4 4: (41) 4 X T 1 T 1 Therefore: ~ 1 ~ ~ K1 ~ + i i i i i 2 2 2 2 2 2 2 _ _ _ _ i=1;i6=2 V =V1 + V2 + V3 = X4 1 1 1 rT r zT z zT z sT 1s T + T + T + T K : (46) 1 2 1 1 3 1 2 4 2 1 1 1 i i i 4 1 1 4 2 2 2 2 2 2 i=1;i6=2 sT 1s + rT Kz + rT Ks + zT s + zT z 2 2 2 1 1 1 1 1 2 1 2 De ne: 4 X T T 1 + sT + sT ~ ^ ~ K^ : (42) = K I + ; = 2I + K I + ; 1 1 2 2 i i i 2 2 2 2 2 M 2 3 M 3 i=1;i6=2 1 By Young inequality, for arbitrary vector a 2 = I + ; 1 = K I + I + ; 4 4 4 1 M 1 Rn1; b 2 Rn1 and diagonal positive de nite matrix H 2 Rnn, the inequality exists as follows: 5 1 = I + ; 2 4 2 T 1 T T a Hb a Ha + b Hb: (43) 4 where i > 0 (i = 2; 3; 4), i > 0 (i = 1; 2). Therefore: 2754 C.G. Li et al./Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2749{2759

_ T D(q) T T J 4. Simulation results and analysis V r1 2r1 z1 3z1 z2 4z2 DM JM The two-link FJ robot shown in Figure 1 with para- 4 metric uncertainties is used to test the feasibility of 1 X T sT s sT s ~ the proposed control method. The dynamic equations 1 1 1 2 2 2 2 i i i=1;i6=2 are derived based on Kane method, and numerical calculation is nished in Simulink of MATLAB. 1 ~ 1 ~T 1 ~ The robot dynamic parameters are given as fol- mii 2 K m22 i 2 2 2 lows:

X4 m1 = 6:07; m2 = 5:76; 1 T 1 T + ii i + 1 1 2 4 T T i=1;i6=2 pc1 = (0:169; 0; 0:0026) ; pc2 = (0:162; 0; 0) ;

1 T 1 T T T + + K ; (47) p01 = (0; 0; 0:055) ; p12 = (0:3; 0; 0:06) : 4 2 2 2 2 2 2 Here, mi (i = 1; 2) is the i-th link mass, and pci (i = where mi is the minimum eigenvalue of i. Select the real number to satisfy the inequality 1; 2) is the mass center of the i-th link relative to the as follows: i-th link coordinate. pi1;i (i = 1; 2) is the relative position between the i-th joint and i 1-th joint. In jjjj jjjj 2 4 addition, 0 < min ; jj3jj; ; jj 1jj; jj 2jj; DM JM " 0:0074 0 0:002 # I = 0 0:0718 0 ; m1 ; m2 ; m3 m4 : (48) c1 2 2 2 2 0:002 0 0:0742 " # Because 1; 2; 1; 2; 3, and 4 are bounded, the in- 0:0065 0 0 equality can be written as follows: Ic2 = 0 0:0654 0 ; 1 X2 1 X4 1 0 0 0:0685 T + T + T K ; (49) 4 i i 2 i i i 2 i i i i=1 i=1;i6=2 J = diag(0:0197; 0:0197); where is the constant. T Eq. (47) can be written according to Eqs. (48) and g = (0; 9:8; 0) ; (49) as follows: K1 = K2 = 100: V_ 2V + : (50) Now, the minimum inertial parameters can be solved Consider the following compact set: as follows: T 2 2 2 1 = (2:84; 0; 0:766; 0:9331; 0; 0:2197) ; 1 := (qd; _qd; qd): jjqdjj + jj _qdj + jjqdjj ; T 2 = (0:01; 0:01) ; 2 := fV pg ; T where p is an arbitrary positive constant. 3 = (100; 100) ; Select =2p. If V = p, V_ 0. Thus, V p T presents an invariant set, i.e., if V (0) p, then V (t) 4 = (0:0197; 0:0197) : p for all t > 0. The ideal trajectories are given as q = sin(2t) and Solving the Inequality Eq. (50) yields: 1d V + V (0) e2t: (51) 2 2 Obviously, all closed-loop signals are uniformly semi- globally bounded, and: lim V (t) : (52) t!1 2

Therefore, by adjusting parameters i (i = 1; 2; 3; 4), i (i = 1; 2), i (i = 1; 2; 3; 4), and i (i = 1; 2; 3; 4) to make bigger, the tracking error will be smaller. Figure 1. 3D model of two-link FJ robot. C.G. Li et al./Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2749{2759 2755

^ T ^ T q2d = sin(2t); the initial positions are speci ed as (1) 1 = (0; 0; 0; 0; 0; 0) , 2 = (0; 0) , ^ T ^ T q1(0) = q2(0) = 0, qm1(0) = qm2(0) = 0, and the 3 = (0; 0) , 4 = (0; 0) , controller parameters are chosen by Eq. (46) as follows: ^ T (2) 1 = (1:42; 0; 0:38; 0:47; 0; 0:11) , ^ T ^ T 1 = 2 = 3 = 4 = 0:0005; 2 = (0:005; 0:005) , 3 = (50; 50) , ^ T 4 = (0:01; 0:01) , 1 = 2 = diag(0:001; 0:001); ^ T (3) 1 = (2:27; 0; 0:61; 0:75; 0; 0:18) , ^ = (0:008; 0:008)T , ^ = (80; 80)T , 1 = 2 = 3 = 4 = diag(0:00005; 0:00005); 2 3 ^ T 4 = (0:016; 0:016) . 1 = diag(60; 60); 2 = diag(60; 60); The tracking errors in the simulation are shown in Figure 3(a) and (b). They reveal that = diag(150; 150); = diag(5; 5): 3 4 the tracking errors of Links 1 and 2 respectively The initial estimated conditions of the inertia param- decrease as the estimated values are closer to ^ T ^ eters are set as: 1(0) = (0; 0; 0; 0; 0; 0) , 2(0) = the nominal value of i, and the their simulation T ^ T ^ T computing times are equal almost. Further, the (0; 0) , 3(0) = (0; 0) , and 4(0) = (0; 0) . The tracking trajectories of Links 1 and 2 are tracking errors can be lower if the estimated values placed in Figure 2(a) and (b) together with their of inertia parameters are closer to the nominal ideal trajectories. The simulation results are shown values. in Figure 2(c) and (d). These gures show that the 2. Uncertainties of parameters of the control laws: For proposed control method results in the very small link verifying the e ectiveness of the proposed adaptive tracking errors (e1 and e2 are both about 5% of the laws, we change these parameters continuously as: ideal trajectories) and ensures the global boundedness of all closed-loop signals with uncertainties. 1 = 2 = 3 = 4= diag(0:00005; 0:00005) sin(t)

1. Uncertainties of the initial values: The initial = = = = 0:0005 sin(t): ^ 1 2 3 4 values of i are not restrained theoretically, that is, they can be selected arbitrarily. For verifying The tracking errors are shown in Figure 4(a) and their e ect, the other parameters should be kept (b), in which e1 and e2 represent the tracking errors ^ unchanged and the initial values of i are set to 0%, of Links 1 and 2, respectively, with the original 50%, and 80%, respectively, of the nominal value i adaptive laws parameters; e11 and e22 represent the as follows: errors with the changed parameters.

Figure 2. The trajectories and tracking errors of Links 1 and 2 with the designed parameters: (a) Ideal and tracking trajectory of Link 1, (b) ideal and tracking trajectory of Link 1, (c) errors of Link 1, and (d) errors of Link 2. 2756 C.G. Li et al./Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2749{2759

^ Figure 3. The tracking errors of Links 1 and 2 with the di erent initial values of i (i = 1; 4): (a) Errors of Link 1 and (b) errors of Link 2.

Figure 4. The tracking errors of Link 1 with di erent parameters of the control laws: (a) Errors of Link 1 with the original and changed parameters and (b) errors of Link 2 with the original and changed parameters.

Figure 5. The tracking errors of Link 1 with di erent external disturbances: (a) Errors of Link 1 without and with external disturbance and (b) errors of Link 2 without and with external disturbance.

It is evident that the tracking errors of Links 1 tracking errors of the links slightly, and tracking errors and 2 are kept small, even though the parameters become weak quickly in 0.1 sec. of adaptive laws are changed. In addition, the errors of links are less than 15% of 3. Uncertainties of external disturbances: Add the desired amplitude in the beginning phase whether the external disturbances on the joints of Links 1 inertia parameters, control laws parameters or external and 2 with functions 50 sin(2t) and 35 sin(4t), disturbance are changed or not. In literature [14], we respectively. investigated an adaptive backstepping control method based on NN; the maximum tracking errors of links The tracking errors are shown in Figure 5(a) and are more than 20% of input amplitude, and time- (b), in which e1 and e2 represent the tracking errors consumption is at least twice what this paper suggests. of Links 1 and 2, respectively, without the external disturbances; e11 and e22 represent the errors with the Conclusions external disturbances. It is clear that the external disturbances cause the In this paper, the uncertainties associated with the C.G. Li et al./Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2749{2759 2757 robot model were estimated by using the inertia pa- J Actuator inertia matrix rameters of FJ robots. The adaptive DSC for FJ robot u Actual control vector was proposed based on backstepping control method, in D ;D Positive constants which the virtual control vectors were derived by using m M C ;G Positive constants the rst-order lter in order to avoid the \explosion M M of derivative". The control vectors and adaptive laws Y(q; _q; q) Regression matrix in the DSC were expressed as the functions of inertia 1 r-dimensional vector of parameters parameters in FJ robots. In addition, the joint tracking r Number of inertia parameters errors could be very insigni cant, even though the a1; v1 Arbitrary n-dimensional vectors initial estimated conditions of the inertia parameters x ; x Functions of state space variables were set to be far away from the real ones, the 1 2 parameters of the control laws were changed, or the Km;KM Positive constants external disturbances were added in the system. Jm;JM Positive constants The simulations of the two-link FJ robot showed qd Desired trajectory of link position that the proposed control system achieved: Q Positive constant 1. Less tracking error, better tracking performance 1; 2 Virtual control vectors and robustness against model uncertainties; 1; 2 Filtering virtual control vectors 2. Simpler calculation to avoid high-order derivative ur Auxiliary control vector and iterative regression matrix; 1; 2; 1 Diagonal positive de nite matrices 3. Shorter computation time to respond; 3; 4; 2; H Diagonal positive de nite matrices 4. Easier selection of control parameters. 1;2 Continuous bounded vectors i Diagonal positive de nite matrix, Moreover, the tracking performance in a short period of i = 1; 2; 3; 4 time can be better if the inertia parameters are selected i Scalar coecients, i > 0, i = 1; 2; 3; 4 precisely. a; b Arbitrary vectors ;^ ;~ Inertia parameters set, estimated Contributors i i i values set, and estimated errors set

Chenggang Li proposed the adaptive dynamic sur- mi Minimum eigenvalue of i face control applied to the exible-joint robot and Real number derived the adaptation laws. Wen Cui carried out Constant the simulation and calculation. Dengdeng Yan studied the control method based on Neural network and p Arbitrary positive constant discussed the uncertainties of FJ robots. Yan Wang mi(i = 1; 2) i-th link mass built the dynamics of robot. Chunming Wang veri ed pci(i = 1; 2) Mass centers of the i-th link relative to the dynamics and simulation in the paper about the the i-th link coordinate adaptive control. pi1;i(i = 1; 2) Relative position between the i-th joint and the i-1-th joint

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27. Liang, T., Wang, W., Wu, S., and Lu, K. \Nonlinear sity in 2013; then, he obtained the MSc degree in attitude control of tiltrotor aircraft based on dynamic Mechanical and Electrical Engineering from Nanjing surface adaptive backstepping", 29th Chinese Control University of Aeronautics and Astronautics, Nanjing in and Decision Conference, Chongqing, China, pp. 603- 2016. His research interests include nonlinear control 608 (2017). and industrial robot design. 28. Yoo, S.J., Jin, B.P., and Choi, Y.H. \Adaptive output feedback control of exible-joint robots using neural Dengdeng Yan received the BSc degree in Mechanical networks: dynamic surface design approach", IEEE Engineering from Zhejiang University of Technology Trans Neural Networks, 19(10), pp. 1712-1726 (2008). in 2015. He is currently pursuing the MSc degree in 29. Ortega, R. and Spong, M.W. \Adaptive motion control Mechanical and Electrical Engineering with Nanjing of rigid robots: a tutorial", Automatica, 25(6), pp. University of Aeronautics and Astronautics, Nanjing. 877-888 (1989). His current research interests include error analysis and compensation of manipulator. Biographies Yan Wang received the BSc degree in Mechanical Chenggang Li received the BSc degree in Hydraulic Engineering from China Three Gorges University in Transmission and Control from Jilin University of 2015. She is currently pursuing the MSc degree in Technology, Changchun in 1998 and obtained the Mechanical and Electrical Engineering with Nanjing MSc and PhD degrees in Mechanical and Electrical University of Aeronautics and Astronautics, Nanjing. Engineering from Jilin University, Changchun, 2001 Her current research interests include structure design and Beijing Institute of Technology, 2004, respectively. and dynamics of manipulator. He is currently an Associate Professor of Mechanical and Electrical Engineering with the Nanjing University Chunming Wang received the BSc degree in Mechan- of Aeronautics and Astronautics. His major research ical Engineering from Nanjing University of Aeronau- activities are system design and industrial robot, tissue tics and Astronautics in 2016. He is currently pursuing modeling and contact analysis, multidimensional sen- the MSc degree in Mechanical and Electrical Engi- sor design, and parallel mechanism. neering with Nanjing University of Aeronautics and Astronautics, Nanjing. His current research interests Wen Cui received the BSc degree in Mechanical and include active compliance control and nonlinear control Electrical Engineering from Nanjing Forestry Univer- of robot.