Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash

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Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash Fanfeng Meng, Lin Zhao∗, Jinpeng Yu School of Automation, Qingdao University, Qingdao 266071, PR China Received 16 January 2019; received in revised form 11 May 2019; accepted 23 June 2019 Available online xxx

Abstract In this paper, the adaptive finite-time command filtered backstepping control is investigated for uncertain manipulator systems with unknown backlash. The finite-time command filters, virtual control signal, adaptive update law and error compensation mechanism are designed respectively, which can deal with the computation complexity problem of conventional backstepping, and guarantee that the joint position tracking error reaches to a sufficiently small region including the origin in finite time although the uncertain parameters and unknown backlash are existed. The dynamic models of a two-link robot manipulator and the PUMA 560 manipulator are applied in simulation to verify the effectiveness of the given control strategy. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Instruction Nowadays, due to the simple structure and easy implementation, robots have gradually been applied to all aspects of industry and life [1–3]. The tracking control problems of robot systems have been the research hot-spot, which require extremely high precision and stability of the control performance [4,5]. Several strategies have been proposed to reach these control goals, such as the computed torque control [6], inverse-dynamics control [7], sliding mode control [8] and sampled-data control [9,10]. Because of the manipulators usually work in ∗

Corresponding author. E-mail address: [email protected] (L. Zhao).

https://doi.org/10.1016/j.jfranklin.2019.06.022 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: F. Meng, L. Zhao and J. Yu, Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash, Journal of the Franklin Institute, https: // doi.org/ 10.1016/ j.jfranklin.2019.06.022

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complex environment, the system parameters for their model cannot be easily obtained. For robot manipulator systems with uncertain parameters and external disturbances, the adaptive control scheme [11] and the disturbance observer [12] are proposed to achieve desired tracking control performances. Although these robot control algorithms have achieved successes in both theory and applications, most of them are merely guaranteed by asymptotic stability [6–13], in which the desired tracking error reaches to zero in infinite time. Recently, the finitetime stability control (FTSC) methods pay close attention to nonlinear control system design. Compared with the asymptotic stability control (ASC) method, the FTSC provides some good properties such as fast response, high tracking precision and strong disturbance rejection ability [14–19]. Hong et al. [20] proposed a homogeneous method based finite-time controller for robot manipulator, but the uncertain parameters were not considered since the homogeneous method could not deal with uncertain systems. The integration of compensation mechanism and terminal sliding mode (TSM) control has been proposed in [21] for air-breathing hypersonic vehicles. The TSM control technique is proposed for manipulators systems in [22,23], which can ensure the system with finite-time convergence although it suffer from influences of system uncertainties and disturbances, but the chattering phenomenon cannot be avoided. The dynamics model of robot manipulator is a strong uncertain nonlinear system. For uncertain nonlinear systems, the dynamic surface control combined with adaptive control technique is seen as an effectively method to solve the tracking problem, which can further avoid the explosion of complexity problem caused by traditional backstepping [24,26–28]. However, how to compensate the filtering errors are not considered in the dynamic surface control, which will increase the difficulty to achieve a better tracking performance. The command filtered backstepping is also proposed for nonlinear control system. At each step of the backstepping, the derivative of virtual signal can be approximated by using the output of command filter, so the terms of computation complex can also be eliminated. Further, by introducing compensation signals in [29,30], the errors caused by command filter can be eliminated. But only the asymptotic convergence rate can be achieved in traditional dynamic surface control and command filtered backstepping control. How to extend the command-filtered backstepping for uncertain manipulator systems such that the closed-loop system can achieve finite-time convergence is remained unsolved, this is one motivation of our research. Moreover, the backlash exists widely in physical systems and devices, for instance, in electro-magnetism, mechanical actuators and manipulator systems. The control system with backlash is an important area of research, and several adaptive control schemes have been proposed for backlash [32–35], but few results are available now for uncertain manipulator systems by considering backlash and finite-time convergence simultaneously. In this paper, a backstepping based adaptive control method will be proposed for manipulator system with uncertain parameters and unknown backlash. The main contributions are stated as follows: (1) Compared with the DSC with asymptotic convergence rate in [24–28], the command filtered backstepping control with asymptotic convergence rate in [29–31] and the adaptive neural network control for manipulator system with backlash in [37], the finitetime convergence is considered in this paper, and a new command filtered backstepping method combined with finite-time control and adaptive control is proposed for manipulator system, which cannot only avoid the explosion of complexity problem for conventional backstepping, but also guarantee the position tracking error converges to a desired neighborhood in finite time although the unknown backlash are existed. Please cite this article as: F. Meng, L. Zhao and J. Yu, Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash, Journal of the Franklin Institute, https: // doi.org/ 10.1016/ j.jfranklin.2019.06.022

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(2) Compared with the homogeneous method based finite-time control for manipulator system in [20], the uncertain parameters are further considered. Compared with the TSM control for manipulator system in [22,23], the established virtual control signal and controller are continuous and can both avoid the chattering phenomenon, which is more adapted to practical applications. The rest of this paper is organized as follows. In Section 2, the system description are given. The control design for manipulator system is given in Section 3. Simulation verification is conducted in Section 4. Section 5 draws a conclusion. 2. System description Considering the following manipulator dynamics model M(q )q¨ + C (q, q˙ )q˙ + G(q ) = ψ

(1)

where q ∈ R , M(q) ∈ R , C (q, q˙ ) ∈ R , G(q ) are the joint configuration vector, symmetric inertia matrix, centripetal and Coriolis torques matrix, gravitational torque, respectively. ψ = φ(τ ) is the joint torque and described by a backlash characteristic, where τ = [τ1 , . . . , τn ]T ∈ Rn , φ(τ ) = [φ1 (τ1 ), . . . , φn (τn )]T ∈ Rn and ⎧ ⎨D(τi (t ) − Sr ), if τ˙i (t ) > 0 and φi (t ) = D(τi (t ) − Sr ) φi (τi ) = D(τi (t ) − Sl ), if τ˙i (t ) < 0 and φi (t ) = D(τi (t ) − Sl ) (2) ⎩ φi (t− ), otherwise n

n×n

n×n

in which D > 0 is the line’s constant slope, Sr > 0 and Sl < 0 are constant parameters, and φ(t− ) implies no change existing in φ i (τ i ). Assumption 1. The backlash parameters are within known bounded ⎧ ⎪ ⎨0 < Dmin ≤ D ≤ Dmax 0 < (DSr )min ≤ DSr ≤ (DSr )max ⎪ ⎩ (DSl )min ≤ DSl ≤ (DSl )max < 0

(3)

Then, we rewrite φ(τ ) as φ(τ ) = Dτ + d (τ ) where d (τ ) = [d1 (τ1 ), . . . , dn (τn )]T , and ⎧ ⎪ ⎨−DSr , if τ˙i (t ) > 0 and φi (t ) = D(τi (t ) − Sr ) di (τi ) = −DSl , if τ˙i (t ) < 0 and φi (t ) = D(τi (t ) − Sl ) ⎪ ⎩ φi (t− ) − Dτi (t ), otherwise

(4)

(5)

Assumption 2. By Eq. (5) and Assumption 1, di (τ i ) satisfies |di (τ i )| ≤ λ, where λ can be selected as λ = max{(DSr )max − (DSl )min }. Remark 1. Note that the D in Eq. (2) represents the slope, Sl and Sr mean the width of nonsmooth backlash nonlinearity, so the boundness assumption of them in Assumption 1 is reasonable. From the definition of di (τ i ) in Eq. (5), we know that di (τ i ) is bounded and the bound λ is not needed in the controller design. The reader is referred to [35–37] for similar assumptions. Please cite this article as: F. Meng, L. Zhao and J. Yu, Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash, Journal of the Franklin Institute, https: // doi.org/ 10.1016/ j.jfranklin.2019.06.022

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The following properties are further given for the dynamics model (1). Property 1. 0 < km1 In < M(q) < km2 In , where km1 , km2 are known positive parameters. Property 2. The dynamics model (1) is linearly parametric with unknown dynamic parameters θ ∈ Rr1 M(q )x˙2 + C (q, q˙ )x2 + G(q ) = Y (q, q˙, x2 , x˙2 )θ

(6)

where Y (q, q˙, x2 , x˙2 ) ∈ Rn×r1 is the dynamic regressor matrix, x2 ∈ Rr1 is differentiable vector and x˙2 is the derivative of x2 .   Property 3. There exists x T M˙ (q ) − 2C (q, q˙ ) x = 0 for any x ∈ Rn . Define the new variables as x1 = q x2 = q˙

(7)

then Eq. (1) can be rewritten as ⎧ ⎪ ⎨x˙1 = x2 M (x1 )x˙2 = −C (x1 , x2 )x2 − G(x1 ) + Dτ + d (τ ) ⎪ ⎩ y = x1

(8)

where y ∈ Rn is the system output. The following Lemmas are given for the proof of the main results in the next section. Lemma 1 [22]. Assume that there exists a continuous positive definite function U(t) which satisfies U˙ + λ1U + λ2U l ≤ 0, ∀t > t0 , where λ1 > 0, λ2 > 0, 0 < l < 1, then U(t) converges to 1−l the equilibrium point in finite time Tr given by Tr ≤ t0 + λ1 (11−l ) ln λ1V λ(t2 0 )+λ2 . Lemma 2 [15]. Let m > 0, n > 0 and δ(x, y) > 0 a real valued function. Then nδ(x, y )− n |x|m+n mδ(x, y )|x|m+n + m+n m+n m

|x|m |y|n ≤

Lemma 3 [15]. For xi ∈ R, i = 1, . . . , N, 0 < q ≤ 1.  N

q  N

q N q 1−q | xi | ≤ | xi | ≤ N | xi | i=1

i=1

(9)

(10)

i=1

Lemma 4 [38]. Considering the finite-time sliding mode differentiator ϕ˙1 = l1 1

l1 = −r1 |ϕ1 − α| 2 sign(ϕ1 − α) + ϕ2 ϕ˙2 = −r2 sign(ϕ2 − l1 )

(11)

where r1 and r2 are positive constants. If input noise satisfies |α − α0 | ≤ κ, then the following inequalities can be satisfied in finite time. |ϕ1 − α0 | ≤ μ1 κ = 1 1

|l1 − α˙ 0 | ≤ λ1 κ 2 = 2

(12)

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where μ1 > 0, λ1 > 0. If the input of Eq. (11) is not influenced by noise, that is α = α0 , then ϕ1 = α0 , l1 = α˙ 0

(13)

can be satisfied in finite-time. 3. Main results The tracking errors are defined as:  x1 = x1 − xd  x2 = x2 − x2,c

(14)

where xd is the reference signal and assume that it with its first-order derivative are known, smooth and bounded. x2,c = [ϕ1,1 , . . . , ϕ1,n ]T is the output of the following sliding mode differentiator and the input is α 1 , where ϕ˙1,z = l1,z 1

l1,z = −r1,z |ϕ1,z − α1,z | 2 sign(ϕ1,z − α1,z ) + ϕ2,z ϕ˙2,z = −r2,z sign(ϕ2,z − l1,z ), z = 1, . . . , n

(15)

Define the virtual control signal α 1 and control torque τ as α1 = −k1 x1 + x˙d − φ1 v1γ τ = −k2 x2 − φ2 v2γ + Y (x1 , x2 , x2,c , x˙2,c )ϑˆ where k1 , k2 , φ 1 , φ 2 are positive constants, and 21 < γ = integers. The updating law of ϑˆ is giving as following ˙ ϑˆ = −2ρ ϑˆ − Y T (x1 , x2 , x2,c , x˙2,c )v2

(16) γ1 γ2

< 1, γ1 > 0, γ2 > 0 are odd

(17)

where ρ > 0, and  is the positive definite symmetric matrix. Define the compensated tracking error signals as v1 = x˜1 − ξ1 , v2 = x˜2

(18)

where ξ 1 is the error compensation formed by ξ˙1 = −k1 ξ1 + x2,c − α1 − κ1 ξ1γ

(19)

with ξ1 (0) = 0. Theorem 1. For the manipulator system (1), the virtual control α 1 and the control torque τ in Eq. (16) with error compensation mechanism (19) and adaptive updating law (17) can be chosen such that the tracking error can converge into the sufficiently small neighborhood of origin in finite time and all signals in the closed-loop system are bounded in finite time. Proof. Choose the Lyapunov function as V1 =

1 T v v1 2 1

(20) 

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Taking the derivative of V1 yields V˙1 = v1T v˙1 = v1T (x˙1 − x˙d − ξ˙1 ) = v1T (α1 + (x2,c − α1 ) + (x2 − x2,c ) − x˙d − ξ˙1 )

(21)

Substituting α 1 and ξ˙1 into Eq. (21) yields V˙1 = v1T (−k1 x˜1 + k1 ξ1 + x˜2 − φ1 v1γ + κ1 ξ1γ ) = v1T (−k1 v1 + v2 − φ1 v1γ + κ1 ξ1γ )

(22)

Choose another Lyapunov function V2 = V1 +

1 T v Mv2 2D 2

(23)

we have v2T M v˙2 vT M˙ v2 + 2 D 2D v2T M(x˙2 − x˙2,c ) v2T M˙ v2 = V˙1 + + D 2D

−Cx2 − G d (τ ) M x˙2,c vT M˙ v2 T + 2 = V˙1 + v2 +τ + − D D D 2D

d (τ ) M x˙2,c vT M˙ v2 T −C(v2 + x2,c ) − G ˙ + 2 = V1 + v2 +τ + − D D D 2D

T ˙ v2 (M − 2C)v2 d (τ ) T −Cx2,c − M x˙2,c − G ˙ = V1 + + v2 +τ + 2D D D

V˙2 = V˙1 +

From Properties 2 and 3, one can write

θ d (τ ) V˙2 = V˙1 + v2T −Y (x1 , x2 , x2,c , x˙2,c ) + τ + D D

(24)

(25)

Define ϑ = Dθ and ϑˆ as the estimation of ϑ. Further define ϑ˜ = ϑ − ϑˆ and substituting τ into Eq. (25) yields V˙2 = − k1 v1T v1 + v1T v2 − φ1 v1T v1γ + κ1 v1T ξ1γ

d (τ ) + v2T −Y (x1 , x2 , x2,c , x˙2,c )ϑ˜ − k2 x˜2 − φ2 v2γ + D

1 T v v2 − φ1 v1T v1γ − φ2 v2T v2γ ≤ − (k1 − 1)v1T v1 − k2 − 2 2 nλ2 + κ1 v1T ξ1γ − v2T Y (x1 , x2 , x2,c , x˙2,c )ϑ˜ + 2 2Dmin ) where v1T v2 ≤ 21 v1T v1 + 21 v2T v2 and v1T d (τ ≤ 21 v1T v1 + D Using Lemma 2, we can obtain

v1T ξ1γ ≤

n s=1

|v1,s ||ξ1,s |γ ≤

nλ2 2 2Dmin

(26) are applied.

n n γ 1 |v1,s |1+γ + |ξ1,s |1+γ 1 + γ s=1 1 + γ s=1

(27)

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Substituting Eq. (27) into Eq. (26), one can write V˙2 ≤ − v2T Y (x1 , x2 , x2,c , x˙2,c )ϑ˜ − (k1 − 1) − φ1 −

κ1 1+γ

Considering V¯ =

n

ξ1T ξ1 , 2

γ v11+ ,s

− φ2

s=1

n



n 1 2 v12,s − k2 − v 2 s=1 2,s s=1

n

1+γ v2,s

s=1

n γ κ1 1+γ nλ2 + ξ1,s + 2 1 + γ s=1 2Dmin

(28)

then we have

V˙¯ = ξ1T ξ˙1 = ξ1T (−k1 ξ1 + x2,c − α1 − κ1 ξ1γ )

(29)

By using Lemma 4, we have x2,c − α1  ≤ 1 in finite-time, then V˙¯ ≤ −k1 ξ1T ξ1 + ξ1 1 − ξ1T ξ11+γ n 1 1 γ ≤ −(k1 − )ξ1T ξ1 + 21 − κ1 ξ11+ ,s 2 2 s=1

(30)

Designing the Lyapunov function V3 = V2 + V¯ , we have

n n 1 2 V˙3 ≤ − v2T Y (x1 , x2 , x2,c , x˙2,c )ϑ˜ − (k1 − 1) v12,s − k2 − v 2 s=1 2,s s=1

n

n n k1 1 2 1+γ 1+γ − φ1 − v − φ2 v2,s − k1 − ξ 1 + γ s=1 1,s 2 s=1 1,s s=1 −

n 2 nλ2 κ1 1+γ ξ1,s + 1 + 2 1 + γ s=1 2 2Dmin

Choosing the Lyapunov function V = V3 +

(31) ϑ˜ T  −1 ϑ˜ . 2

Using Eq. (17), one can write

n n 1 2 V˙ ≤ − v2T Y (x1 , x2 , x2,c , x˙2,c )ϑ˜ − (k1 − 1) v12,s − k2 − v 2 s=1 2,s s=1

n

n n κ1 1 2 1+γ 1+γ − φ1 − v − φ2 v2,s − κ1 − ξ 1 + γ s=1 1,s 2 s=1 1,s s=1 n 2 nλ2 κ1 1+γ ˙ ξ1,s + 1 + − ϑ˜ T  −1 ϑˆ 2 1 + γ s=1 2 2Dmin

n n 1 2 2 ≤ − (k1 − 1) v1,s − k2 − v 2 s=1 2 s=1

n

n n κ1 1 2 γ 1+γ − φ1 − v11+ − φ v − κ − ξ 2 1 2,s 1 + γ s=1 ,s 2 s=1 1,s s=1

−

r1 n 21 nλ2 κ1 1+γ − ξ + + +2 ρ ϑ˜s ϑˆs 2 1 + γ s=1 1,s 2 2Dmin s=1

(32)

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Then, ρ ϑ˜s ϑˆs ≤ − ρ(2−1) ϑ˜s2 + ρ ϑ 2 ,  > 21 , we get 2 2 s

n n 1 2 2 ˙ V ≤ − (k1 − 1) v1,s − k2 − v 2 s=1 2,s s=1

n

n n κ1 1 2 γ 1+γ − φ1 − v11+ − φ v − k − ξ 2 1 2,s 1 + γ s=1 ,s 2 s=1 1,s s=1 r1 n 1+γ 21 nλ2 κ1 1+γ − ξ + + − (ς ϑ˜s2 ) 2 2 1 + γ s=1 1,s 2 2Dmin s=1

+

r1

(ς ϑ˜s2 )

1+γ 2

−

s=1

r1

2ς ϑ˜s2 +

s=1

r1

ρϑs2

(33)

s=1

1+γ where ς = ρ(2−1) . If ς ϑ˜s2 < 1, we get (ς ϑ˜s2 )( 2 ) − ς ϑ˜s2 + ρϑs2 < 1 − ς ϑ˜s2 + ρϑs2 < 1 + 2 1+γ ρϑs2 If ς ϑ˜s2 ≥ 1, we get (ς ϑ˜s2 )( 2 ) − ς ϑ˜s2 + ρϑs2 ≤ ς ϑ˜s2 − ς ϑ˜s2 + ρϑs2 = ρϑs2 . Then, we obtain

n n 1 2 V˙ ≤ − (k1 − 1) v12,s − k2 − v 2 s=1 2,s s=1

n

n n κ1 1 2 γ 1+γ − φ1 − v11+ − φ v − k − ξ 2 1 2,s 1 + γ s=1 ,s 2 s=1 1,s s=1

− −

r1 n 1+γ 2 nλ2 κ1 1+γ ξ1,s + 1 + − (ςs ϑ˜s2 ) 2 2 1 + γ s=1 2 2Dmin s=1 r1

ςs ϑ˜s2 +

r1

s=1

(1 + ρϑs2 )

(34)

s=1

Base on Lemma 3, we have V˙ ≤ −1V − 2V

1+γ 2

+ 3

(35)

where

  Dmin (2k2 − 1) 2ς , 1 = min 2k1 − 2, , , 2κ − 1 1 km2 λmax ( −1 ) 

1+γ

1+2 γ 

1+γ 1+γ 2Dmin 2 2ς κ1 κ1 2 2 , φ2 2 2 , 2 = min φ1 − , , 1+γ km2 1+γ λmax ( −1 ) 1 21 nλ2 + + (1 + ρϑs2 ) 2 2 2Dmin s=1

r

3 =

Then, one can write V˙ ≤ −1V − 2V

1+γ 2

− (1 − )2V

1+γ 2

+ 3

(36)

or V˙ ≤ −1V − (1 − )1V − 2V

1+γ 2

+ 3

(37)

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1+γ 1+γ 3 where 0 <  < 1. If V 2 > (1−) , one can rewrite Eq. (36) as V˙ ≤ −1V − 2V 2 , 2 2 3 so v1 , ξ1 , ϑ˜ converge to the region (v1 , ξ1 , ϑ˜ ) ∈ {(V ≤ ( (1−) ) 1+γ } in finite time T ≤ 2

1 1 (1− 1+2 γ )

2 V

1+γ 2

ln

1V 1−

1+γ 2

(0)+2 . 2

If V >

3 , (1−)1

one can rewrite Eq. (37) as V˙ ≤ −1V −

, thus from Lemma 1, we know v1 , ξ1 , ϑ˜ converge to the region (v1 , ξ1 , ϑ˜ ) ∈ {V ≤

3 } (1−)1

in finite time, and the setting time is T ≤ 

T¯ = max

1 1 (1 −

1+γ 2

)

ln

1V 1−

1+γ 2

2

(0) + 2

1 1 (1− 1+2 γ )

,

ln

1V 1−

1 1 (1 −

1+γ 2

)

ln

1+γ 2

2

(0)+2

1 V 1 −

. Then, denote

1+γ 2

(0) + 2 2



(38) v1 can finally reach to the region ⎧ ⎫ 

1+2 γ ⎬ ⎨ 3 23 v1  ≤ min , 2 ⎩ (1 − )1 ⎭ (1 − )2 in finite time T¯ . ξ 1 can finally reach to the region ⎧ ⎫ 

1+2 γ ⎬ ⎨ 3 23 ξ1  ≤ min , 2 ⎩ (1 − )1 ⎭ (1 − )2 in finite time T¯ . So x˜1  will finally reach to the region ⎧  ⎫ 

1+2 γ ⎬ ⎨ 3 23 x˜1  ≤ min 2 ,2 2 ⎩ (1 − )1 ⎭ (1 − )2

(39)

(40)

(41)

in finite time T¯ . Remark 2. Note that the virtual control signal α 1 in Eq. (16) is used to guarantee the first subsystem of Eq. (8) with the desired performance at first step of backstepping. However, the derivative of α 1 is needed in traditional backstepping design, which will result in computation complex problem [24,26–28]. We can use the output of command filter (15) to replace the computation of α˙ 1 in the second step of backstepping and eliminate the computation problem. But the filtering error (x2,c − α1 ) is existed before the filter (15) achieves stability, which will affect control qualities. The error compensation mechanism in Eq. (19) is further used to remove the filtering error and guarantee the desired control performance. Remark 3. To achieve Theorem 1, the control parameters k1 , k2 , φ 1 , κ 1 should satisfy k1 > 1, k2 >

1 1 κ1 , κ 1 > , φ1 > 2 2 1+γ

(42)

Form Eq. (41), we can see that the radius of region of tracking error is decided by the parameters 1 and 2 , and larger 1 and 2 can get smaller radiuses of convergence region. From the definition of 1 and 2 under Eq. (35), we know that larger k1 , k2 , κ 1 ,  can get κ1 larger 1 , and larger φ 1 , φ 2 , κ 1 ,  can get larger 2 . From the term φ1 − 1+ , we know γ γ κ1 γ κ1 κ1 that if we choose φ1 = κ1 , φ1 − 1+γ = 1+γ , so larger κ 1 can guarantee larger 1+γ , and then Please cite this article as: F. Meng, L. Zhao and J. Yu, Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash, Journal of the Franklin Institute, https: // doi.org/ 10.1016/ j.jfranklin.2019.06.022

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guarantee larger 2 . Further, the setting time T¯ is also decided by the parameters 1 and 2 , so we can also choose larger k1 , k2 , φ 1 , φ 2 , κ 1 ,  to guarantee shorter convergence time. 4. Simulation results Example 1. We consider the dynamic model for two-link robot manipulator, in which M(q) = [Mmn ] ∈ R2×2 and C(q, q˙ ) = [Cmn ] ∈ R2×2 are given by M11 = a1 + 2a2 cos(q2 ), M21 = M12 = a3 + a2 cos(q2 ), M22 = a3 , C11 = −a2 sin (q2 )q˙2 , C12 = −a2 sin (q2 )(q˙1 + q˙2 ), C21 = a2 sin (q2 )q˙1 , C22 = 0, and 2 2 2 a1 = I1 + m1 Lc1 + m2 L12 + I2 + m2 Lc2 , a2 = m2 L1 Lc2 , a3 = I2 + m2 Lc2

with the links’ masses m1 and m2 , the inertia’s moments I1 and I2 , the links’ lengths L1 and L2 and the links’ mass centers Lc1 and Lc2 . We assume that the gravitational torque G(q) = 0 for simplicity. Let x2,c = (x2,c,1 , x2,c,2 ) ∈ R2 . For Property 2, the parameter vector is given by θ = (a1 , a2 , a3 )T ∈ R3 and the regressor matrix Y (q, q˙, x2,c , x˙2,c ) = [Ymn ] ∈ R2×3 with Y11 = x˙2,c,1 , Y12 = 2 cos(q2 )x˙2,c,1 + cos(q2 )x˙2,c,2 − sin (q2 )q˙2 x2,c,1 − sin (q2 )(q˙1 + q˙2 )x2,c,2 , Y13 = x˙2,c,2 , Y21 = 0, Y22 = cos(q2 )x˙2,c,1 + sin (q2 )q˙1 x2,c,1 , Y23 = x˙2,c,1 + x˙2,c,2 The parameters of the manipulator are chosen as I1 = 0.52 kg m2 , I2 = 0.41 kg m2 , m1 = 1.6 kg, m2 = 1.7 kg, L1 = 2 m, L2 = 1.8 m, Lc1 = 1.1 m, Lc2 = 1.2 m with initial condition q1 (0) = 0.43π rad, q2 (0) = −0.52π rad, q˙1 (0) = 0 rad/s, q˙2 (0) = 0 rad/s, and ϑˆ (0) = (1.6, 1.2, 2)T . The reference signal is given by xd (t ) = (2 ∗ sin (t ), 2 ∗ cos(t ))T rad. The control parameters are chosen as γ = 35 , k1 = 10, k2 = 10, φ1 = 10, φ2 = 10, κ1 = 10, r1,z = 30, r2,z = 30(z = 1, 2) and  = diag{0.1, 0.1, 0.1}, ρ = 1. The parameters for the backlash are chosen as D = 1, Sr = 1, Sl = −1. The curves of qz and xd,z (z = 1, 2) under the proposed control algorithm are shown in Fig. 1, and the curves of ϑˆz (z = 1, 2, 3) are shown in Fig. 2. It can be seen that the joint configuration vector of manipulator ultimately converges to the reference signal in finite-time with sufficient accuracy, which means that the desired tracking error is guaranteed although the uncertain parameters and unknown backlash existed. The curves of α 1,z and x2,c,z (z = 1, 2) are shown in Fig. 3, which show that the virtual signals are fast filtered under the given finite-time command filters. To further demonstrate the effectiveness of proposed scheme, we compare the given algorithm with traditional command filtered backstepping, which means that the convergence Please cite this article as: F. Meng, L. Zhao and J. Yu, Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash, Journal of the Franklin Institute, https: // doi.org/ 10.1016/ j.jfranklin.2019.06.022

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reduces to the asymptotic convergence, and the control parameters are given by different cases: I. k1 = 10, k2 = 10, φ1 = 0, φ2 = 0, κ1 = 0, r1,z = 30, r2,z = 30(z = 1, 2); II. k1 = 15, k2 = 15, φ1 = 0, φ2 = 0, κ1 = 0, r1,z = 30, r2,z = 30(z = 1, 2); III. k1 = 20, k2 = 20, φ1 = 0, φ2 = 0, κ1 = 0, r1,z = 30, r2,z = 30(z = 1, 2). The curves of qz and xd,z (z = 1, 2) under the three group parameters under the command filtered backstepping algorithm are shown in Figs. 4–6. To compare the tracking control performances, the overall tracking error OTE = x˜1  is applied. The curves of OTE under the finite-time command filtered backstepping and the traditional command filtered backstepping are given in Fig. 7. It can be seen that the proposed algorithm cannot only guarantee the closed-loop system has fast convergence performance and but also ensure the closed-loop system has better tracking performance. Example 2. We further consider the numerical simulations of a PUMA 560 manipulator to demonstrate the effectiveness of the proposed algorithm. The PUMA 560 manipulator is widely used in both industry and academic, and the dynamic parameters for this robot arm have been identified and documented in [39]. For the PUAM 560 manipulator, the key point of its control is its big-arm and small-arm, so in order to simplify, only the big-arm and small-arm are considered in this example, which means that the model is simplified from 6 DOF to 2 DOF. The reference signal is given by xd (t ) = (2 ∗ sin (t ), 2 ∗ cos(t ))T rad. The control parameters are chosen as γ = 35 , k1 = 50, k2 = 50, φ1 = 50, φ2 = 50, κ1 = 50, r1,z = 80, r2,z = 80(z = 1, 2) and  = diag{0.1, 0.1, 0.1}, ρ = 1. The curves of qz and xd,z (z = 1, 2) under the proposed control algorithm are shown in Fig. 8. It can also be seen that the joint configuration vector of manipulator ultimately converges to the reference Please cite this article as: F. Meng, L. Zhao and J. Yu, Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash, Journal of the Franklin Institute, https: // doi.org/ 10.1016/ j.jfranklin.2019.06.022

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signal in finite-time with sufficient accuracy. Further, as in Example 1, we compare the given algorithm with traditional command filtered backstepping, and the control parameters are given by different cases: I. k1 = 50, k2 = 50, φ1 = 0, φ2 = 0, κ1 = 0, r1,z = 80, r2,z = 80(z = 1, 2); II. k1 = 80, k2 = 80, φ1 = 0, φ2 = 0, κ1 = 0, r1,z = 80, r2,z = 80(z = 1, 2); III. k1 = 100, k2 = 100, φ1 = 0, φ2 = 0, κ1 = 0, r1,z = 80, r2,z = 80(z = 1, 2). The curves of OTE under the finite-time command filtered backstepping and the traditional command filtered backstepping are given in Fig. 9. It can be seen that the similar performances are achieved as in Example 1. 5. Conclusions This paper proposes a new adaptive finite-time command filtered backstepping algorithm for manipulator system with uncertain parameters and unknown backlash. The proposed scheme cannot only avoid the explosion of complexity problem for conventional backstepping, but also guarantee the position tracking error converges to a desired neighborhood in finite time although the uncertain parameters and unknown backlash existed. Moreover, the used sliding mode differentiator and error compensation mechanism can make a better control quality. Since the continuous virtual signal and controller are designed, the chattering phenomenon can be avoided. Acknowledgments This work was supported by the National Natural Science Foundation of China (61603204 and 61573204), the Shandong Province Outstanding Youth Fund (ZR2018JL020), the Qingdao Application Basic Research Project (16-5-1-22-jch) and the Taishan Scholar Special Project Fund (TSQN20161026). References [1] C. Ishii, T. Shen, K. Tamura, Robust model-following control for a robot manipulator, IEE Proc. Control Theory Appl. 144 (1) (1997) 53–60. [2] Z.J. Yang, Y. Fukushima, P. Qin, Decentralized adaptive robust control of robot manipulators using disturbance observers, IEEE Trans. Control Syst. Technol. 20 (5) (2012) 1357–1365. [3] M.J. Cai, Z.R. Xiang, Adaptive finite-time fault-tolerant consensus protocols for multiple mechanical systems, J. Frankl. Inst. 353 (6) (2016) 1386–1408. [4] P.R. Ouyang, W.J. Zhang, M.M. Gupta, An adaptive switching learning control method for trajectory tracking of robot manipulators, Mechatronics 16 (1) (2006) 51–61. [5] H. Wang, Consensus of networked mechanical systems with communication delays: a unified framework, IEEE Trans. Autom. Control 59 (6) (2014) 1571–1576. [6] Z.S. Song, J.Q. Yi, D.B. Zhao, X.C. Li, A computed torque controller for uncertain robotic manipulator systems: fuzzy approach, Fuzzy Sets Syst. 154 (2) (2005) 208–226. [7] H. Asada, Z.D. Ma, H. Tokumaru, Inverse dynamics of flexible robot arms: modeling and computation for trajectory control, J. Dyn. Syst. Meas. Control 112 (2) (1990) 177–185. [8] S.H. Huh, Z. Bien, Robust sliding mode control of a robot manipulator based on variable structure-model reference adaptive control approach, IET Control Theory Appl. 1 (5) (2007) 1355–1363. [9] W.B. Zhang, Y. Tang, T. Huang, A.V. Vasilakos, Consensus of networked Euler–Lagrange systems under time– varying sampled-data control, IEEE Trans. Ind. Inform. 14 (2) (2018) 535–544. [10] W.B. Zhang, Q.L. Han, Y. Tang, Y.R. Liu, Sampled-data control for a class of linear time-varying systems, Automatica 103 (2019) 126–134. [11] Z.J. Yang, Y. Fukushima, P. Qin, Decentralized adaptive robust control of robot manipulators using disturbance observers, IEEE Trans. Control Syst. Technol. 20 (5) (2012) 1357–1365. Please cite this article as: F. Meng, L. Zhao and J. Yu, Backstepping based adaptive finite-time tracking control of manipulator systems with uncertain parameters and unknown backlash, Journal of the Franklin Institute, https: // doi.org/ 10.1016/ j.jfranklin.2019.06.022

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